Properties

Label 12-2520e6-1.1-c1e6-0-4
Degree $12$
Conductor $2.561\times 10^{20}$
Sign $1$
Analytic cond. $6.63843\times 10^{7}$
Root an. cond. $4.48578$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·11-s − 8·19-s − 5·25-s + 6·29-s + 20·31-s − 36·41-s − 3·49-s + 12·59-s + 48·61-s − 8·71-s − 34·79-s + 8·101-s − 10·109-s + 65·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 4.22·11-s − 1.83·19-s − 25-s + 1.11·29-s + 3.59·31-s − 5.62·41-s − 3/7·49-s + 1.56·59-s + 6.14·61-s − 0.949·71-s − 3.82·79-s + 0.796·101-s − 0.957·109-s + 5.90·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(6.63843\times 10^{7}\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2520} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1876658601\)
\(L(\frac12)\) \(\approx\) \(0.1876658601\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + p T^{2} - 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} \)
7 \( ( 1 + T^{2} )^{3} \)
good11 \( ( 1 + 7 T + 41 T^{2} + 146 T^{3} + 41 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 9 T^{2} + 491 T^{4} - 2882 T^{6} + 491 p^{2} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 53 T^{2} + 1539 T^{4} - 31118 T^{6} + 1539 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 + 4 T + 43 T^{2} + 160 T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 2 p T^{2} + 1887 T^{4} - 43972 T^{6} + 1887 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 3 T + 15 T^{2} - 66 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 - 10 T + 101 T^{2} - 540 T^{3} + 101 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{3} \)
41 \( ( 1 + 18 T + 191 T^{2} + 1388 T^{3} + 191 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 178 T^{2} + 15575 T^{4} - 836124 T^{6} + 15575 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 105 T^{2} + 6339 T^{4} - 285798 T^{6} + 6339 p^{2} T^{8} - 105 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 130 T^{2} + 13655 T^{4} - 792060 T^{6} + 13655 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 - 6 T + 99 T^{2} - 752 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 24 T + 365 T^{2} - 3368 T^{3} + 365 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 174 T^{2} + 20567 T^{4} - 1533188 T^{6} + 20567 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + 4 T + 193 T^{2} + 504 T^{3} + 193 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 346 T^{2} + 55487 T^{4} - 5172972 T^{6} + 55487 p^{2} T^{8} - 346 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 17 T + 205 T^{2} + 2138 T^{3} + 205 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 70 T^{2} + 1179 T^{4} + 304148 T^{6} + 1179 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 95 T^{2} - 464 T^{3} + 95 p T^{4} + p^{3} T^{6} )^{2} \)
97 \( 1 - 469 T^{2} + 98339 T^{4} - 12075534 T^{6} + 98339 p^{2} T^{8} - 469 p^{4} T^{10} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.65858354855003893101126239931, −4.59994488451215588399093542221, −4.48972827846643921426873194349, −4.23427601888781306023596922337, −3.91392252897854704702633376272, −3.81465500942225348979093730084, −3.67839226278887463688795582420, −3.66231536049378816869838850409, −3.52523163992515637473555000619, −2.99053629160952327448134668438, −2.96440028540264394697127446812, −2.91335398132730255943807104845, −2.85139326443831076600829781629, −2.57674070095778207485601798572, −2.31992878004315136132630275793, −2.27923785208909139433265285932, −2.20520732190493527744107221225, −2.03132308992885941311253924239, −1.81732444337859393287451200494, −1.31537123417937857453737081584, −1.23616658409360555570265336177, −1.11373875470825605196781497375, −0.67034462645585601602404197236, −0.23966721023130268378553724670, −0.11401191124837913187552011752, 0.11401191124837913187552011752, 0.23966721023130268378553724670, 0.67034462645585601602404197236, 1.11373875470825605196781497375, 1.23616658409360555570265336177, 1.31537123417937857453737081584, 1.81732444337859393287451200494, 2.03132308992885941311253924239, 2.20520732190493527744107221225, 2.27923785208909139433265285932, 2.31992878004315136132630275793, 2.57674070095778207485601798572, 2.85139326443831076600829781629, 2.91335398132730255943807104845, 2.96440028540264394697127446812, 2.99053629160952327448134668438, 3.52523163992515637473555000619, 3.66231536049378816869838850409, 3.67839226278887463688795582420, 3.81465500942225348979093730084, 3.91392252897854704702633376272, 4.23427601888781306023596922337, 4.48972827846643921426873194349, 4.59994488451215588399093542221, 4.65858354855003893101126239931

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

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