Properties

Label 24-570e12-1.1-c1e12-0-0
Degree $24$
Conductor $1.176\times 10^{33}$
Sign $1$
Analytic cond. $7.90357\times 10^{7}$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s + 3·9-s − 4·11-s + 3·16-s + 6·19-s − 25-s − 16·29-s + 9·36-s + 10·41-s − 12·44-s + 14·49-s + 8·59-s − 28·61-s − 2·64-s + 44·71-s + 18·76-s − 36·79-s + 3·81-s + 6·89-s − 12·99-s − 3·100-s + 4·101-s + 8·109-s − 48·116-s − 70·121-s + 32·125-s + 127-s + ⋯
L(s)  = 1  + 3/2·4-s + 9-s − 1.20·11-s + 3/4·16-s + 1.37·19-s − 1/5·25-s − 2.97·29-s + 3/2·36-s + 1.56·41-s − 1.80·44-s + 2·49-s + 1.04·59-s − 3.58·61-s − 1/4·64-s + 5.22·71-s + 2.06·76-s − 4.05·79-s + 1/3·81-s + 0.635·89-s − 1.20·99-s − 0.299·100-s + 0.398·101-s + 0.766·109-s − 4.45·116-s − 6.36·121-s + 2.86·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(7.90357\times 10^{7}\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.082559425\)
\(L(\frac12)\) \(\approx\) \(3.082559425\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{3} \)
3 \( ( 1 - T^{2} + T^{4} )^{3} \)
5 \( 1 + T^{2} - 32 T^{3} + 6 T^{4} - 16 T^{5} + 501 T^{6} - 16 p T^{7} + 6 p^{2} T^{8} - 32 p^{3} T^{9} + p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 - 3 T + 24 T^{2} - 25 T^{3} + 24 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
good7 \( ( 1 - p T^{2} + 50 T^{4} - 531 T^{6} + 50 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} )^{2} \)
11 \( ( 1 + T + 20 T^{2} + 17 T^{3} + 20 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{4} \)
13 \( 1 + 66 T^{2} + 2413 T^{4} + 63062 T^{6} + 99698 p T^{8} + 21859066 T^{10} + 309202565 T^{12} + 21859066 p^{2} T^{14} + 99698 p^{5} T^{16} + 63062 p^{6} T^{18} + 2413 p^{8} T^{20} + 66 p^{10} T^{22} + p^{12} T^{24} \)
17 \( 1 + 58 T^{2} + 1421 T^{4} + 32966 T^{6} + 901610 T^{8} + 17162838 T^{10} + 265402029 T^{12} + 17162838 p^{2} T^{14} + 901610 p^{4} T^{16} + 32966 p^{6} T^{18} + 1421 p^{8} T^{20} + 58 p^{10} T^{22} + p^{12} T^{24} \)
23 \( 1 - 57 T^{2} + 1051 T^{4} - 14620 T^{6} + 514481 T^{8} - 5020283 T^{10} - 107097658 T^{12} - 5020283 p^{2} T^{14} + 514481 p^{4} T^{16} - 14620 p^{6} T^{18} + 1051 p^{8} T^{20} - 57 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 + 8 T - 7 T^{2} - 100 T^{3} + 310 T^{4} - 2502 T^{5} - 39679 T^{6} - 2502 p T^{7} + 310 p^{2} T^{8} - 100 p^{3} T^{9} - 7 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
31 \( ( 1 + 77 T^{2} + 16 T^{3} + 77 p T^{4} + p^{3} T^{6} )^{4} \)
37 \( ( 1 - 195 T^{2} + 16646 T^{4} - 799199 T^{6} + 16646 p^{2} T^{8} - 195 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
41 \( ( 1 - 5 T - 27 T^{2} + 838 T^{3} - 53 p T^{4} - 15633 T^{5} + 286354 T^{6} - 15633 p T^{7} - 53 p^{3} T^{8} + 838 p^{3} T^{9} - 27 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
43 \( 1 + 178 T^{2} + 16301 T^{4} + 1091270 T^{6} + 61619066 T^{8} + 3149941242 T^{10} + 144853393557 T^{12} + 3149941242 p^{2} T^{14} + 61619066 p^{4} T^{16} + 1091270 p^{6} T^{18} + 16301 p^{8} T^{20} + 178 p^{10} T^{22} + p^{12} T^{24} \)
47 \( 1 + 18 T^{2} - 3035 T^{4} - 58474 T^{6} + 2792426 T^{8} + 20466154 T^{10} - 1410027091 T^{12} + 20466154 p^{2} T^{14} + 2792426 p^{4} T^{16} - 58474 p^{6} T^{18} - 3035 p^{8} T^{20} + 18 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 + 111 T^{2} + 6051 T^{4} + 54868 T^{6} - 13880691 T^{8} - 1024261299 T^{10} - 60566886642 T^{12} - 1024261299 p^{2} T^{14} - 13880691 p^{4} T^{16} + 54868 p^{6} T^{18} + 6051 p^{8} T^{20} + 111 p^{10} T^{22} + p^{12} T^{24} \)
59 \( ( 1 - 4 T - 73 T^{2} - 316 T^{3} + 2590 T^{4} + 26304 T^{5} - 111877 T^{6} + 26304 p T^{7} + 2590 p^{2} T^{8} - 316 p^{3} T^{9} - 73 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 14 T + 31 T^{2} - 550 T^{3} - 2860 T^{4} + 6224 T^{5} + 108821 T^{6} + 6224 p T^{7} - 2860 p^{2} T^{8} - 550 p^{3} T^{9} + 31 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 194 T^{2} + 11805 T^{4} + 1009630 T^{6} + 163131554 T^{8} + 10601096334 T^{10} + 430434923317 T^{12} + 10601096334 p^{2} T^{14} + 163131554 p^{4} T^{16} + 1009630 p^{6} T^{18} + 11805 p^{8} T^{20} + 194 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 - 22 T + 133 T^{2} - 1062 T^{3} + 25020 T^{4} - 183208 T^{5} + 617607 T^{6} - 183208 p T^{7} + 25020 p^{2} T^{8} - 1062 p^{3} T^{9} + 133 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
73 \( 1 + 258 T^{2} + 33409 T^{4} + 2866118 T^{6} + 188835542 T^{8} + 10449018838 T^{10} + 642415211525 T^{12} + 10449018838 p^{2} T^{14} + 188835542 p^{4} T^{16} + 2866118 p^{6} T^{18} + 33409 p^{8} T^{20} + 258 p^{10} T^{22} + p^{12} T^{24} \)
79 \( ( 1 + 18 T - 5 T^{2} + 26 T^{3} + 29186 T^{4} + 146554 T^{5} - 896341 T^{6} + 146554 p T^{7} + 29186 p^{2} T^{8} + 26 p^{3} T^{9} - 5 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( ( 1 - 370 T^{2} + 63911 T^{4} - 6638556 T^{6} + 63911 p^{2} T^{8} - 370 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 3 T + 19 T^{2} - 2968 T^{3} + 5487 T^{4} - 44093 T^{5} + 3576750 T^{6} - 44093 p T^{7} + 5487 p^{2} T^{8} - 2968 p^{3} T^{9} + 19 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( 1 + 382 T^{2} + 74889 T^{4} + 10425050 T^{6} + 1185931790 T^{8} + 116693583282 T^{10} + 11078552578789 T^{12} + 116693583282 p^{2} T^{14} + 1185931790 p^{4} T^{16} + 10425050 p^{6} T^{18} + 74889 p^{8} T^{20} + 382 p^{10} T^{22} + p^{12} T^{24} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.55287046100137211936146618332, −3.44164166226445083775426457609, −3.37003789005943331196079045226, −3.30009393588390727619051794219, −3.18784699242236344491500367619, −2.85934769513652608281318332075, −2.83204230618248165218034641128, −2.71176097914444366493079247070, −2.64778911571576412377538435661, −2.58219799097758715240057134992, −2.44382145506491122270150127240, −2.37640281161383854380499734481, −2.22706662608221637647831393893, −2.17480481632563877963301792906, −1.94415917968314343127077539860, −1.92629532654439267617418932374, −1.74774865446199518683536122956, −1.49714096434320371940017866081, −1.46141729252297520155499701817, −1.31318759789367771285183895512, −1.12906673731350389457720736591, −0.936612609738759809173212533708, −0.915549630782262319049072312143, −0.37730676226889435342019738786, −0.19167503216436158948729416464, 0.19167503216436158948729416464, 0.37730676226889435342019738786, 0.915549630782262319049072312143, 0.936612609738759809173212533708, 1.12906673731350389457720736591, 1.31318759789367771285183895512, 1.46141729252297520155499701817, 1.49714096434320371940017866081, 1.74774865446199518683536122956, 1.92629532654439267617418932374, 1.94415917968314343127077539860, 2.17480481632563877963301792906, 2.22706662608221637647831393893, 2.37640281161383854380499734481, 2.44382145506491122270150127240, 2.58219799097758715240057134992, 2.64778911571576412377538435661, 2.71176097914444366493079247070, 2.83204230618248165218034641128, 2.85934769513652608281318332075, 3.18784699242236344491500367619, 3.30009393588390727619051794219, 3.37003789005943331196079045226, 3.44164166226445083775426457609, 3.55287046100137211936146618332

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

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