Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11 \cdot 13^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 2·5-s − 7-s + 3·8-s + 2·10-s − 11-s + 14-s − 16-s − 2·17-s − 4·19-s + 2·20-s + 22-s − 25-s + 28-s + 2·29-s − 8·31-s − 5·32-s + 2·34-s + 2·35-s − 6·37-s + 4·38-s − 6·40-s + 10·41-s − 4·43-s + 44-s − 8·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s − 0.377·7-s + 1.06·8-s + 0.632·10-s − 0.301·11-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.447·20-s + 0.213·22-s − 1/5·25-s + 0.188·28-s + 0.371·29-s − 1.43·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s − 0.986·37-s + 0.648·38-s − 0.948·40-s + 1.56·41-s − 0.609·43-s + 0.150·44-s − 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(117117\)    =    \(3^{2} \cdot 7 \cdot 11 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{117117} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 117117,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.08502608892844, −13.53002870242102, −12.99653556687224, −12.70705699200681, −12.19104342154552, −11.55846610520125, −10.93641403672176, −10.76632077081068, −10.14429942028252, −9.600716152704235, −9.093391259983253, −8.759074427710435, −8.124523055891399, −7.770455529078601, −7.369653042139906, −6.621861169621392, −6.264936360758524, −5.355530812442643, −4.915913386846132, −4.352635840734566, −3.719311869198687, −3.463643645703611, −2.432443796407193, −1.868076817433288, −1.016605726297617, 0, 0, 1.016605726297617, 1.868076817433288, 2.432443796407193, 3.463643645703611, 3.719311869198687, 4.352635840734566, 4.915913386846132, 5.355530812442643, 6.264936360758524, 6.621861169621392, 7.369653042139906, 7.770455529078601, 8.124523055891399, 8.759074427710435, 9.093391259983253, 9.600716152704235, 10.14429942028252, 10.76632077081068, 10.93641403672176, 11.55846610520125, 12.19104342154552, 12.70705699200681, 12.99653556687224, 13.53002870242102, 14.08502608892844

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