# Properties

 Label 24-637e12-1.1-c1e12-0-5 Degree $24$ Conductor $4.463\times 10^{33}$ Sign $1$ Analytic cond. $2.99915\times 10^{8}$ Root an. cond. $2.25532$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·3-s − 4·4-s + 13·9-s − 12·11-s − 12·12-s + 2·13-s + 8·16-s − 17·17-s + 9·19-s + 3·23-s + 35·25-s + 32·27-s − 29-s − 36·33-s − 52·36-s − 15·37-s + 6·39-s + 6·41-s + 11·43-s + 48·44-s + 24·48-s − 51·51-s − 8·52-s + 16·53-s + 27·57-s + 27·59-s − 5·61-s + ⋯
 L(s)  = 1 + 1.73·3-s − 2·4-s + 13/3·9-s − 3.61·11-s − 3.46·12-s + 0.554·13-s + 2·16-s − 4.12·17-s + 2.06·19-s + 0.625·23-s + 7·25-s + 6.15·27-s − 0.185·29-s − 6.26·33-s − 8.66·36-s − 2.46·37-s + 0.960·39-s + 0.937·41-s + 1.67·43-s + 7.23·44-s + 3.46·48-s − 7.14·51-s − 1.10·52-s + 2.19·53-s + 3.57·57-s + 3.51·59-s − 0.640·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$24$$ Conductor: $$7^{24} \cdot 13^{12}$$ Sign: $1$ Analytic conductor: $$2.99915\times 10^{8}$$ Root analytic conductor: $$2.25532$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{637} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(24,\ 7^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.464344205$$ $$L(\frac12)$$ $$\approx$$ $$2.464344205$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1$$
13 $$1 - 2 T - 18 T^{2} + 17 T^{3} + 341 T^{4} + 63 T^{5} - 6395 T^{6} + 63 p T^{7} + 341 p^{2} T^{8} + 17 p^{3} T^{9} - 18 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}$$
good2 $$1 + p^{2} T^{2} + p^{3} T^{4} + 5 T^{6} - p^{4} T^{8} - 3 p^{4} T^{10} + 3 p^{2} T^{11} - 111 T^{12} + 3 p^{3} T^{13} - 3 p^{6} T^{14} - p^{8} T^{16} + 5 p^{6} T^{18} + p^{11} T^{20} + p^{12} T^{22} + p^{12} T^{24}$$
3 $$1 - p T - 4 T^{2} + 19 T^{3} + p T^{4} - 16 p T^{5} - 19 T^{6} + 47 p T^{7} + 22 T^{8} - 73 p T^{9} - 95 p T^{10} - 124 T^{11} + 2353 T^{12} - 124 p T^{13} - 95 p^{3} T^{14} - 73 p^{4} T^{15} + 22 p^{4} T^{16} + 47 p^{6} T^{17} - 19 p^{6} T^{18} - 16 p^{8} T^{19} + p^{9} T^{20} + 19 p^{9} T^{21} - 4 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24}$$
5 $$1 - 7 p T^{2} + 601 T^{4} - 6779 T^{6} + 56873 T^{8} - 379271 T^{10} + 2080181 T^{12} - 379271 p^{2} T^{14} + 56873 p^{4} T^{16} - 6779 p^{6} T^{18} + 601 p^{8} T^{20} - 7 p^{11} T^{22} + p^{12} T^{24}$$
11 $$1 + 12 T + 107 T^{2} + 708 T^{3} + 3961 T^{4} + 18831 T^{5} + 79826 T^{6} + 305121 T^{7} + 98539 p T^{8} + 3659247 T^{9} + 11994110 T^{10} + 39258729 T^{11} + 129081947 T^{12} + 39258729 p T^{13} + 11994110 p^{2} T^{14} + 3659247 p^{3} T^{15} + 98539 p^{5} T^{16} + 305121 p^{5} T^{17} + 79826 p^{6} T^{18} + 18831 p^{7} T^{19} + 3961 p^{8} T^{20} + 708 p^{9} T^{21} + 107 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24}$$
17 $$1 + p T + 91 T^{2} + 80 T^{3} + 354 T^{4} + 13150 T^{5} + 66387 T^{6} + 80838 T^{7} + 350534 T^{8} + 5285257 T^{9} + 24796664 T^{10} + 64859774 T^{11} + 191483227 T^{12} + 64859774 p T^{13} + 24796664 p^{2} T^{14} + 5285257 p^{3} T^{15} + 350534 p^{4} T^{16} + 80838 p^{5} T^{17} + 66387 p^{6} T^{18} + 13150 p^{7} T^{19} + 354 p^{8} T^{20} + 80 p^{9} T^{21} + 91 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24}$$
19 $$1 - 9 T + 115 T^{2} - 792 T^{3} + 6216 T^{4} - 34521 T^{5} + 202721 T^{6} - 930966 T^{7} + 4436762 T^{8} - 17205552 T^{9} + 72759135 T^{10} - 265082310 T^{11} + 1203810589 T^{12} - 265082310 p T^{13} + 72759135 p^{2} T^{14} - 17205552 p^{3} T^{15} + 4436762 p^{4} T^{16} - 930966 p^{5} T^{17} + 202721 p^{6} T^{18} - 34521 p^{7} T^{19} + 6216 p^{8} T^{20} - 792 p^{9} T^{21} + 115 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24}$$
23 $$1 - 3 T - 79 T^{2} - 4 p T^{3} + 4336 T^{4} + 12687 T^{5} - 111893 T^{6} - 708984 T^{7} + 1416466 T^{8} + 17335200 T^{9} + 27721271 T^{10} - 199047772 T^{11} - 1123052895 T^{12} - 199047772 p T^{13} + 27721271 p^{2} T^{14} + 17335200 p^{3} T^{15} + 1416466 p^{4} T^{16} - 708984 p^{5} T^{17} - 111893 p^{6} T^{18} + 12687 p^{7} T^{19} + 4336 p^{8} T^{20} - 4 p^{10} T^{21} - 79 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24}$$
29 $$1 + T - 3 p T^{2} + 178 T^{3} + 128 p T^{4} - 15194 T^{5} - 71197 T^{6} + 410532 T^{7} + 447076 T^{8} - 2082223 T^{9} - 28038240 T^{10} - 61539492 T^{11} + 1484469159 T^{12} - 61539492 p T^{13} - 28038240 p^{2} T^{14} - 2082223 p^{3} T^{15} + 447076 p^{4} T^{16} + 410532 p^{5} T^{17} - 71197 p^{6} T^{18} - 15194 p^{7} T^{19} + 128 p^{9} T^{20} + 178 p^{9} T^{21} - 3 p^{11} T^{22} + p^{11} T^{23} + p^{12} T^{24}$$
31 $$1 - 140 T^{2} + 9930 T^{4} - 450715 T^{6} + 15153149 T^{8} - 429707637 T^{10} + 12513324193 T^{12} - 429707637 p^{2} T^{14} + 15153149 p^{4} T^{16} - 450715 p^{6} T^{18} + 9930 p^{8} T^{20} - 140 p^{10} T^{22} + p^{12} T^{24}$$
37 $$1 + 15 T + 261 T^{2} + 2790 T^{3} + 30330 T^{4} + 259743 T^{5} + 2192045 T^{6} + 15890796 T^{7} + 113579334 T^{8} + 728456778 T^{9} + 4731329259 T^{10} + 28450921716 T^{11} + 178599894873 T^{12} + 28450921716 p T^{13} + 4731329259 p^{2} T^{14} + 728456778 p^{3} T^{15} + 113579334 p^{4} T^{16} + 15890796 p^{5} T^{17} + 2192045 p^{6} T^{18} + 259743 p^{7} T^{19} + 30330 p^{8} T^{20} + 2790 p^{9} T^{21} + 261 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24}$$
41 $$1 - 6 T + 87 T^{2} - 450 T^{3} + 3253 T^{4} - 22749 T^{5} + 119000 T^{6} - 655563 T^{7} + 2153315 T^{8} + 1902885 T^{9} - 21527678 T^{10} + 258610773 T^{11} - 1564835407 T^{12} + 258610773 p T^{13} - 21527678 p^{2} T^{14} + 1902885 p^{3} T^{15} + 2153315 p^{4} T^{16} - 655563 p^{5} T^{17} + 119000 p^{6} T^{18} - 22749 p^{7} T^{19} + 3253 p^{8} T^{20} - 450 p^{9} T^{21} + 87 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24}$$
43 $$1 - 11 T - 88 T^{2} + 799 T^{3} + 8712 T^{4} - 35151 T^{5} - 615381 T^{6} + 1493311 T^{7} + 26239709 T^{8} - 29370632 T^{9} - 1062722582 T^{10} - 167088622 T^{11} + 50469301069 T^{12} - 167088622 p T^{13} - 1062722582 p^{2} T^{14} - 29370632 p^{3} T^{15} + 26239709 p^{4} T^{16} + 1493311 p^{5} T^{17} - 615381 p^{6} T^{18} - 35151 p^{7} T^{19} + 8712 p^{8} T^{20} + 799 p^{9} T^{21} - 88 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24}$$
47 $$1 - 523 T^{2} + 127033 T^{4} - 18954871 T^{6} + 1935563261 T^{8} - 142521847195 T^{10} + 7760213706737 T^{12} - 142521847195 p^{2} T^{14} + 1935563261 p^{4} T^{16} - 18954871 p^{6} T^{18} + 127033 p^{8} T^{20} - 523 p^{10} T^{22} + p^{12} T^{24}$$
53 $$( 1 - 8 T + 280 T^{2} - 1716 T^{3} + 33468 T^{4} - 160748 T^{5} + 2272305 T^{6} - 160748 p T^{7} + 33468 p^{2} T^{8} - 1716 p^{3} T^{9} + 280 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
59 $$1 - 27 T + 442 T^{2} - 5373 T^{3} + 47243 T^{4} - 314460 T^{5} + 1356131 T^{6} - 82953 T^{7} - 42334024 T^{8} + 284989455 T^{9} + 1152047535 T^{10} - 39755713086 T^{11} + 391751656497 T^{12} - 39755713086 p T^{13} + 1152047535 p^{2} T^{14} + 284989455 p^{3} T^{15} - 42334024 p^{4} T^{16} - 82953 p^{5} T^{17} + 1356131 p^{6} T^{18} - 314460 p^{7} T^{19} + 47243 p^{8} T^{20} - 5373 p^{9} T^{21} + 442 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24}$$
61 $$1 + 5 T - 266 T^{2} - 887 T^{3} + 40211 T^{4} + 80704 T^{5} - 4493895 T^{6} - 5470553 T^{7} + 403359860 T^{8} + 265514337 T^{9} - 30729415795 T^{10} - 6069628418 T^{11} + 2020503763113 T^{12} - 6069628418 p T^{13} - 30729415795 p^{2} T^{14} + 265514337 p^{3} T^{15} + 403359860 p^{4} T^{16} - 5470553 p^{5} T^{17} - 4493895 p^{6} T^{18} + 80704 p^{7} T^{19} + 40211 p^{8} T^{20} - 887 p^{9} T^{21} - 266 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24}$$
67 $$1 + 15 T + 295 T^{2} + 3300 T^{3} + 37385 T^{4} + 337086 T^{5} + 2917472 T^{6} + 22840887 T^{7} + 168746996 T^{8} + 1188097515 T^{9} + 6974328225 T^{10} + 52016179137 T^{11} + 290088466863 T^{12} + 52016179137 p T^{13} + 6974328225 p^{2} T^{14} + 1188097515 p^{3} T^{15} + 168746996 p^{4} T^{16} + 22840887 p^{5} T^{17} + 2917472 p^{6} T^{18} + 337086 p^{7} T^{19} + 37385 p^{8} T^{20} + 3300 p^{9} T^{21} + 295 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24}$$
71 $$1 - 30 T + 653 T^{2} - 10590 T^{3} + 145585 T^{4} - 1771113 T^{5} + 19362440 T^{6} - 197718039 T^{7} + 1886667311 T^{8} - 17315970285 T^{9} + 153782228042 T^{10} - 1330750537803 T^{11} + 11347613768747 T^{12} - 1330750537803 p T^{13} + 153782228042 p^{2} T^{14} - 17315970285 p^{3} T^{15} + 1886667311 p^{4} T^{16} - 197718039 p^{5} T^{17} + 19362440 p^{6} T^{18} - 1771113 p^{7} T^{19} + 145585 p^{8} T^{20} - 10590 p^{9} T^{21} + 653 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24}$$
73 $$1 - 476 T^{2} + 113546 T^{4} - 17894407 T^{6} + 2099846477 T^{8} - 197467415277 T^{10} + 15594902689281 T^{12} - 197467415277 p^{2} T^{14} + 2099846477 p^{4} T^{16} - 17894407 p^{6} T^{18} + 113546 p^{8} T^{20} - 476 p^{10} T^{22} + p^{12} T^{24}$$
79 $$( 1 - 35 T + 818 T^{2} - 13321 T^{3} + 181646 T^{4} - 2028165 T^{5} + 19730991 T^{6} - 2028165 p T^{7} + 181646 p^{2} T^{8} - 13321 p^{3} T^{9} + 818 p^{4} T^{10} - 35 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
83 $$1 - 533 T^{2} + 130220 T^{4} - 19632136 T^{6} + 2123453318 T^{8} - 188332543809 T^{10} + 15642556571895 T^{12} - 188332543809 p^{2} T^{14} + 2123453318 p^{4} T^{16} - 19632136 p^{6} T^{18} + 130220 p^{8} T^{20} - 533 p^{10} T^{22} + p^{12} T^{24}$$
89 $$1 - 48 T + 1471 T^{2} - 33744 T^{3} + 649840 T^{4} - 10865652 T^{5} + 162607993 T^{6} - 2211532644 T^{7} + 27724965467 T^{8} - 323293945128 T^{9} + 3530567707708 T^{10} - 36297047904324 T^{11} + 352202680354553 T^{12} - 36297047904324 p T^{13} + 3530567707708 p^{2} T^{14} - 323293945128 p^{3} T^{15} + 27724965467 p^{4} T^{16} - 2211532644 p^{5} T^{17} + 162607993 p^{6} T^{18} - 10865652 p^{7} T^{19} + 649840 p^{8} T^{20} - 33744 p^{9} T^{21} + 1471 p^{10} T^{22} - 48 p^{11} T^{23} + p^{12} T^{24}$$
97 $$1 - 3 T + 406 T^{2} - 1209 T^{3} + 93212 T^{4} - 404427 T^{5} + 13886771 T^{6} - 91101243 T^{7} + 1491993611 T^{8} - 15578221158 T^{9} + 128844085584 T^{10} - 1984216560294 T^{11} + 11260776987123 T^{12} - 1984216560294 p T^{13} + 128844085584 p^{2} T^{14} - 15578221158 p^{3} T^{15} + 1491993611 p^{4} T^{16} - 91101243 p^{5} T^{17} + 13886771 p^{6} T^{18} - 404427 p^{7} T^{19} + 93212 p^{8} T^{20} - 1209 p^{9} T^{21} + 406 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$