Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.43·2-s + 3.90·4-s − 1.43·5-s − 7-s − 4.64·8-s + 3.47·10-s + 2.30·13-s + 2.43·14-s + 3.46·16-s − 1.14·17-s + 5.47·19-s − 5.59·20-s − 3.00·23-s − 2.95·25-s − 5.59·26-s − 3.90·28-s + 3.16·29-s − 6.99·31-s + 0.861·32-s + 2.78·34-s + 1.43·35-s − 8.16·37-s − 13.3·38-s + 6.64·40-s − 5.59·41-s + 10.5·43-s + 7.30·46-s + ⋯
L(s)  = 1  − 1.71·2-s + 1.95·4-s − 0.639·5-s − 0.377·7-s − 1.64·8-s + 1.09·10-s + 0.638·13-s + 0.649·14-s + 0.866·16-s − 0.277·17-s + 1.25·19-s − 1.25·20-s − 0.626·23-s − 0.590·25-s − 1.09·26-s − 0.738·28-s + 0.588·29-s − 1.25·31-s + 0.152·32-s + 0.476·34-s + 0.241·35-s − 1.34·37-s − 2.16·38-s + 1.05·40-s − 0.873·41-s + 1.61·43-s + 1.07·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7623} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5217684010$
$L(\frac12)$  $\approx$  $0.5217684010$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + 2.43T + 2T^{2} \)
5 \( 1 + 1.43T + 5T^{2} \)
13 \( 1 - 2.30T + 13T^{2} \)
17 \( 1 + 1.14T + 17T^{2} \)
19 \( 1 - 5.47T + 19T^{2} \)
23 \( 1 + 3.00T + 23T^{2} \)
29 \( 1 - 3.16T + 29T^{2} \)
31 \( 1 + 6.99T + 31T^{2} \)
37 \( 1 + 8.16T + 37T^{2} \)
41 \( 1 + 5.59T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 + 9.28T + 53T^{2} \)
59 \( 1 - 6.89T + 59T^{2} \)
61 \( 1 + 8.50T + 61T^{2} \)
67 \( 1 - 7.61T + 67T^{2} \)
71 \( 1 - 1.92T + 71T^{2} \)
73 \( 1 - 4.83T + 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + 9.40T + 83T^{2} \)
89 \( 1 - 4.35T + 89T^{2} \)
97 \( 1 + 6.56T + 97T^{2} \)
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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

−7.917382364682249213458417672477, −7.42858880310223811514609863497, −6.84251800854045768419079400227, −6.07076663609294732899623002093, −5.28131912055786420394527595784, −4.05476586550508274327281657198, −3.39249599567391088803258740906, −2.39634956127631973810816390912, −1.46549675322359722394018249135, −0.48999723273947608286955237481, 0.48999723273947608286955237481, 1.46549675322359722394018249135, 2.39634956127631973810816390912, 3.39249599567391088803258740906, 4.05476586550508274327281657198, 5.28131912055786420394527595784, 6.07076663609294732899623002093, 6.84251800854045768419079400227, 7.42858880310223811514609863497, 7.917382364682249213458417672477

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