Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 11-s + 2·13-s + 4·17-s − 6·19-s − 25-s + 8·29-s − 8·31-s + 10·37-s + 8·41-s + 2·43-s + 8·47-s + 2·53-s + 2·55-s − 12·59-s − 10·61-s + 4·65-s − 12·67-s + 8·71-s − 6·73-s + 2·79-s − 16·83-s + 8·85-s − 14·89-s − 12·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.301·11-s + 0.554·13-s + 0.970·17-s − 1.37·19-s − 1/5·25-s + 1.48·29-s − 1.43·31-s + 1.64·37-s + 1.24·41-s + 0.304·43-s + 1.16·47-s + 0.274·53-s + 0.269·55-s − 1.56·59-s − 1.28·61-s + 0.496·65-s − 1.46·67-s + 0.949·71-s − 0.702·73-s + 0.225·79-s − 1.75·83-s + 0.867·85-s − 1.48·89-s − 1.23·95-s + 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(77616\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{77616} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 77616,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−14.18988415139971, −13.85092646650329, −13.32111313994153, −12.74115248599868, −12.40705039329489, −11.90124165750013, −11.05382491975315, −10.84030094732727, −10.28162226887667, −9.717446589462990, −9.254198319333990, −8.821813429956403, −8.201130206568439, −7.649685480556141, −7.112568650479981, −6.333268449881420, −5.937223888702586, −5.731240679515176, −4.797456227540916, −4.238388656081238, −3.768328792255835, −2.796233625021225, −2.488962314310414, −1.529011894761435, −1.145536077234290, 0, 1.145536077234290, 1.529011894761435, 2.488962314310414, 2.796233625021225, 3.768328792255835, 4.238388656081238, 4.797456227540916, 5.731240679515176, 5.937223888702586, 6.333268449881420, 7.112568650479981, 7.649685480556141, 8.201130206568439, 8.821813429956403, 9.254198319333990, 9.717446589462990, 10.28162226887667, 10.84030094732727, 11.05382491975315, 11.90124165750013, 12.40705039329489, 12.74115248599868, 13.32111313994153, 13.85092646650329, 14.18988415139971

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