Properties

Label 12-735e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.577\times 10^{17}$
Sign $1$
Analytic cond. $40868.3$
Root an. cond. $2.42260$
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  + 4-s − 2·5-s − 3·9-s + 12·11-s + 2·16-s + 12·19-s − 2·20-s + 25-s − 4·29-s − 4·31-s − 3·36-s − 4·41-s + 12·44-s + 6·45-s − 24·55-s + 32·59-s + 12·61-s + 6·64-s + 12·71-s + 12·76-s − 24·79-s − 4·80-s + 6·81-s + 28·89-s − 24·95-s − 36·99-s + 100-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s − 9-s + 3.61·11-s + 1/2·16-s + 2.75·19-s − 0.447·20-s + 1/5·25-s − 0.742·29-s − 0.718·31-s − 1/2·36-s − 0.624·41-s + 1.80·44-s + 0.894·45-s − 3.23·55-s + 4.16·59-s + 1.53·61-s + 3/4·64-s + 1.42·71-s + 1.37·76-s − 2.70·79-s − 0.447·80-s + 2/3·81-s + 2.96·89-s − 2.46·95-s − 3.61·99-s + 1/10·100-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(40868.3\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{735} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{6} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.271140225\)
\(L(\frac12)\) \(\approx\) \(7.271140225\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T^{2} )^{3} \)
5 \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 \)
good2 \( 1 - T^{2} - T^{4} - 3 T^{6} - p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \)
11 \( ( 1 - 2 T + p T^{2} )^{6} \)
13 \( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 359 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 70 T^{2} + 2415 T^{4} - 51220 T^{6} + 2415 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 - 6 T + 53 T^{2} - 188 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 106 T^{2} + 5183 T^{4} - 150348 T^{6} + 5183 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 + 2 T + 41 T^{2} - 60 T^{3} + 41 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 46 T^{2} + 1399 T^{4} - 74788 T^{6} + 1399 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 + 2 T + 63 T^{2} - 36 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 2839 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 154 T^{2} + 12143 T^{4} - 652332 T^{6} + 12143 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 146 T^{2} + 14103 T^{4} - 884828 T^{6} + 14103 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 - 16 T + 113 T^{2} - 608 T^{3} + 113 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 6 T + 131 T^{2} - 484 T^{3} + 131 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 274 T^{2} + 36103 T^{4} - 2962972 T^{6} + 36103 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 2 T + p T^{2} )^{6} \)
73 \( 1 - 298 T^{2} + 43775 T^{4} - 3982284 T^{6} + 43775 p^{2} T^{8} - 298 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 12 T + 221 T^{2} + 1576 T^{3} + 221 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 319 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 8719 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.41850729321703544272927685129, −5.37098360459983443326350085548, −5.34936345412940930464020021887, −5.13050816174725436104763815235, −4.99884007345760852462677783950, −4.49077159180770507744289603237, −4.48649515361515054101412048689, −4.37110249655078203849110105381, −4.04515006699795042465510280448, −3.85275289790133415522046864631, −3.65395471218706518316633281896, −3.63547783436169013411225716614, −3.48850403083860805689671673557, −3.46127311182856594064911783111, −3.02123399643056820234641464575, −2.93781293254516387576425200299, −2.69326002907762983036432274868, −2.20685070228205092138121302023, −2.19643238624277705314844881968, −1.86718664373272650919099690877, −1.67309995943286332352979678029, −1.33517352106343975310830010203, −0.974767415738005954941835462056, −0.76204998513013853300185557497, −0.62562193882476938977424344886, 0.62562193882476938977424344886, 0.76204998513013853300185557497, 0.974767415738005954941835462056, 1.33517352106343975310830010203, 1.67309995943286332352979678029, 1.86718664373272650919099690877, 2.19643238624277705314844881968, 2.20685070228205092138121302023, 2.69326002907762983036432274868, 2.93781293254516387576425200299, 3.02123399643056820234641464575, 3.46127311182856594064911783111, 3.48850403083860805689671673557, 3.63547783436169013411225716614, 3.65395471218706518316633281896, 3.85275289790133415522046864631, 4.04515006699795042465510280448, 4.37110249655078203849110105381, 4.48649515361515054101412048689, 4.49077159180770507744289603237, 4.99884007345760852462677783950, 5.13050816174725436104763815235, 5.34936345412940930464020021887, 5.37098360459983443326350085548, 5.41850729321703544272927685129

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

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