Properties

Label 14-4600e7-1.1-c1e7-0-1
Degree $14$
Conductor $4.358\times 10^{25}$
Sign $-1$
Analytic cond. $9.02078\times 10^{10}$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $7$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 4·7-s − 5·9-s − 7·11-s − 7·13-s − 7·19-s + 12·21-s − 7·23-s − 27·27-s − 11·29-s − 10·31-s − 21·33-s − 19·37-s − 21·39-s − 16·41-s + 6·43-s + 6·47-s − 25·49-s − 15·53-s − 21·57-s − 11·59-s + 5·61-s − 20·63-s + 9·67-s − 21·69-s − 14·71-s − 10·73-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.51·7-s − 5/3·9-s − 2.11·11-s − 1.94·13-s − 1.60·19-s + 2.61·21-s − 1.45·23-s − 5.19·27-s − 2.04·29-s − 1.79·31-s − 3.65·33-s − 3.12·37-s − 3.36·39-s − 2.49·41-s + 0.914·43-s + 0.875·47-s − 3.57·49-s − 2.06·53-s − 2.78·57-s − 1.43·59-s + 0.640·61-s − 2.51·63-s + 1.09·67-s − 2.52·69-s − 1.66·71-s − 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{14} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{14} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(2^{21} \cdot 5^{14} \cdot 23^{7}\)
Sign: $-1$
Analytic conductor: \(9.02078\times 10^{10}\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(7\)
Selberg data: \((14,\ 2^{21} \cdot 5^{14} \cdot 23^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( ( 1 + T )^{7} \)
good3 \( 1 - p T + 14 T^{2} - 10 p T^{3} + 85 T^{4} - 152 T^{5} + 341 T^{6} - 536 T^{7} + 341 p T^{8} - 152 p^{2} T^{9} + 85 p^{3} T^{10} - 10 p^{5} T^{11} + 14 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \)
7 \( 1 - 4 T + 41 T^{2} - 127 T^{3} + 759 T^{4} - 1908 T^{5} + 8307 T^{6} - 16918 T^{7} + 8307 p T^{8} - 1908 p^{2} T^{9} + 759 p^{3} T^{10} - 127 p^{4} T^{11} + 41 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 + 7 T + 50 T^{2} + 24 p T^{3} + 1304 T^{4} + 5217 T^{5} + 20319 T^{6} + 69648 T^{7} + 20319 p T^{8} + 5217 p^{2} T^{9} + 1304 p^{3} T^{10} + 24 p^{5} T^{11} + 50 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 + 7 T + 6 p T^{2} + 454 T^{3} + 2833 T^{4} + 13096 T^{5} + 59795 T^{6} + 217800 T^{7} + 59795 p T^{8} + 13096 p^{2} T^{9} + 2833 p^{3} T^{10} + 454 p^{4} T^{11} + 6 p^{6} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 + 61 T^{2} + 105 T^{3} + 1641 T^{4} + 6090 T^{5} + 29753 T^{6} + 146422 T^{7} + 29753 p T^{8} + 6090 p^{2} T^{9} + 1641 p^{3} T^{10} + 105 p^{4} T^{11} + 61 p^{5} T^{12} + p^{7} T^{14} \)
19 \( 1 + 7 T + 80 T^{2} + 560 T^{3} + 3576 T^{4} + 20921 T^{5} + 103617 T^{6} + 488640 T^{7} + 103617 p T^{8} + 20921 p^{2} T^{9} + 3576 p^{3} T^{10} + 560 p^{4} T^{11} + 80 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 11 T + 147 T^{2} + 984 T^{3} + 8570 T^{4} + 46128 T^{5} + 336468 T^{6} + 1566138 T^{7} + 336468 p T^{8} + 46128 p^{2} T^{9} + 8570 p^{3} T^{10} + 984 p^{4} T^{11} + 147 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 + 10 T + 107 T^{2} + 617 T^{3} + 5296 T^{4} + 28550 T^{5} + 208730 T^{6} + 954795 T^{7} + 208730 p T^{8} + 28550 p^{2} T^{9} + 5296 p^{3} T^{10} + 617 p^{4} T^{11} + 107 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 + 19 T + 327 T^{2} + 3518 T^{3} + 35497 T^{4} + 275617 T^{5} + 2069855 T^{6} + 12731020 T^{7} + 2069855 p T^{8} + 275617 p^{2} T^{9} + 35497 p^{3} T^{10} + 3518 p^{4} T^{11} + 327 p^{5} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 + 16 T + 299 T^{2} + 2981 T^{3} + 33474 T^{4} + 254854 T^{5} + 2163504 T^{6} + 13191531 T^{7} + 2163504 p T^{8} + 254854 p^{2} T^{9} + 33474 p^{3} T^{10} + 2981 p^{4} T^{11} + 299 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 - 6 T + 271 T^{2} - 1484 T^{3} + 32567 T^{4} - 155570 T^{5} + 2251961 T^{6} - 8845208 T^{7} + 2251961 p T^{8} - 155570 p^{2} T^{9} + 32567 p^{3} T^{10} - 1484 p^{4} T^{11} + 271 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 6 T + 151 T^{2} - 562 T^{3} + 11256 T^{4} - 558 p T^{5} + 624278 T^{6} - 1222452 T^{7} + 624278 p T^{8} - 558 p^{3} T^{9} + 11256 p^{3} T^{10} - 562 p^{4} T^{11} + 151 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + 15 T + 311 T^{2} + 3174 T^{3} + 40385 T^{4} + 331769 T^{5} + 3249663 T^{6} + 21958644 T^{7} + 3249663 p T^{8} + 331769 p^{2} T^{9} + 40385 p^{3} T^{10} + 3174 p^{4} T^{11} + 311 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 + 11 T + 161 T^{2} + 1606 T^{3} + 15241 T^{4} + 104989 T^{5} + 902577 T^{6} + 6212660 T^{7} + 902577 p T^{8} + 104989 p^{2} T^{9} + 15241 p^{3} T^{10} + 1606 p^{4} T^{11} + 161 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 - 5 T + 112 T^{2} + 218 T^{3} + 7820 T^{4} + 11169 T^{5} + 779059 T^{6} + 139788 T^{7} + 779059 p T^{8} + 11169 p^{2} T^{9} + 7820 p^{3} T^{10} + 218 p^{4} T^{11} + 112 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 - 9 T + 225 T^{2} - 1886 T^{3} + 24401 T^{4} - 213987 T^{5} + 1845009 T^{6} - 16692652 T^{7} + 1845009 p T^{8} - 213987 p^{2} T^{9} + 24401 p^{3} T^{10} - 1886 p^{4} T^{11} + 225 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 + 14 T + 341 T^{2} + 3253 T^{3} + 50270 T^{4} + 415786 T^{5} + 5172636 T^{6} + 36898691 T^{7} + 5172636 p T^{8} + 415786 p^{2} T^{9} + 50270 p^{3} T^{10} + 3253 p^{4} T^{11} + 341 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 + 10 T + 481 T^{2} + 3914 T^{3} + 100568 T^{4} + 665794 T^{5} + 163422 p T^{6} + 63270868 T^{7} + 163422 p^{2} T^{8} + 665794 p^{2} T^{9} + 100568 p^{3} T^{10} + 3914 p^{4} T^{11} + 481 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 + 32 T + 719 T^{2} + 12292 T^{3} + 173907 T^{4} + 2116176 T^{5} + 22653621 T^{6} + 212954872 T^{7} + 22653621 p T^{8} + 2116176 p^{2} T^{9} + 173907 p^{3} T^{10} + 12292 p^{4} T^{11} + 719 p^{5} T^{12} + 32 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 - T + 359 T^{2} + 572 T^{3} + 58259 T^{4} + 231357 T^{5} + 6074581 T^{6} + 29368688 T^{7} + 6074581 p T^{8} + 231357 p^{2} T^{9} + 58259 p^{3} T^{10} + 572 p^{4} T^{11} + 359 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 + 24 T + 669 T^{2} + 10556 T^{3} + 175975 T^{4} + 2095656 T^{5} + 25712747 T^{6} + 239948584 T^{7} + 25712747 p T^{8} + 2095656 p^{2} T^{9} + 175975 p^{3} T^{10} + 10556 p^{4} T^{11} + 669 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 7 T + 310 T^{2} - 3110 T^{3} + 64344 T^{4} - 560137 T^{5} + 9041429 T^{6} - 66958876 T^{7} + 9041429 p T^{8} - 560137 p^{2} T^{9} + 64344 p^{3} T^{10} - 3110 p^{4} T^{11} + 310 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.20931925617068692541836080379, −4.16839834395885472374634820473, −3.91199960886521535300995804731, −3.75071408261960247271556006558, −3.53639025723533032283837588392, −3.52878356067452232782938261618, −3.43665249332236604756316086873, −3.32010419563285426020089737470, −3.12878812474692061998713221762, −3.08080063616651354388454507220, −3.07740331263286469319310028460, −2.75457097100686176368292047757, −2.56676407170049838835689753677, −2.56013718914314383544108047239, −2.52562473655515477203955694368, −2.23758334698004661215997212622, −2.22648227851314190355287558004, −2.17801451802281865148522756127, −1.87274060247329682452644795993, −1.85346072833896978509969072517, −1.70675689675953100502713882154, −1.36886688188199271753749348557, −1.32379314990075453694803910124, −1.32324582688478178892996734115, −1.18736980023198134194016845642, 0, 0, 0, 0, 0, 0, 0, 1.18736980023198134194016845642, 1.32324582688478178892996734115, 1.32379314990075453694803910124, 1.36886688188199271753749348557, 1.70675689675953100502713882154, 1.85346072833896978509969072517, 1.87274060247329682452644795993, 2.17801451802281865148522756127, 2.22648227851314190355287558004, 2.23758334698004661215997212622, 2.52562473655515477203955694368, 2.56013718914314383544108047239, 2.56676407170049838835689753677, 2.75457097100686176368292047757, 3.07740331263286469319310028460, 3.08080063616651354388454507220, 3.12878812474692061998713221762, 3.32010419563285426020089737470, 3.43665249332236604756316086873, 3.52878356067452232782938261618, 3.53639025723533032283837588392, 3.75071408261960247271556006558, 3.91199960886521535300995804731, 4.16839834395885472374634820473, 4.20931925617068692541836080379

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.