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  − 2.60·2-s + 4.78·4-s − 4.31·5-s − 7-s − 7.23·8-s + 11.2·10-s + 2.46·13-s + 2.60·14-s + 9.29·16-s + 1.18·17-s + 3.42·19-s − 20.6·20-s + 8.32·23-s + 13.6·25-s − 6.41·26-s − 4.78·28-s − 1.45·29-s + 2.98·31-s − 9.71·32-s − 3.08·34-s + 4.31·35-s + 6.34·37-s − 8.92·38-s + 31.2·40-s + 6.16·41-s − 3.15·43-s − 21.6·46-s + ⋯
L(s)  = 1  − 1.84·2-s + 2.39·4-s − 1.93·5-s − 0.377·7-s − 2.55·8-s + 3.55·10-s + 0.683·13-s + 0.695·14-s + 2.32·16-s + 0.287·17-s + 0.786·19-s − 4.61·20-s + 1.73·23-s + 2.73·25-s − 1.25·26-s − 0.903·28-s − 0.269·29-s + 0.536·31-s − 1.71·32-s − 0.529·34-s + 0.729·35-s + 1.04·37-s − 1.44·38-s + 4.94·40-s + 0.962·41-s − 0.480·43-s − 3.19·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5809860862$
$L(\frac12)$  $\approx$  $0.5809860862$
$L(\frac{3}{2})$   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 \( 1 + 2.60T + 2T^{2} \)
5 \( 1 + 4.31T + 5T^{2} \)
13 \( 1 - 2.46T + 13T^{2} \)
17 \( 1 - 1.18T + 17T^{2} \)
19 \( 1 - 3.42T + 19T^{2} \)
23 \( 1 - 8.32T + 23T^{2} \)
29 \( 1 + 1.45T + 29T^{2} \)
31 \( 1 - 2.98T + 31T^{2} \)
37 \( 1 - 6.34T + 37T^{2} \)
41 \( 1 - 6.16T + 41T^{2} \)
43 \( 1 + 3.15T + 43T^{2} \)
47 \( 1 - 8.73T + 47T^{2} \)
53 \( 1 + 6.68T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 7.76T + 61T^{2} \)
67 \( 1 - 3.11T + 67T^{2} \)
71 \( 1 - 5.48T + 71T^{2} \)
73 \( 1 + 8.66T + 73T^{2} \)
79 \( 1 - 8.83T + 79T^{2} \)
83 \( 1 - 9.66T + 83T^{2} \)
89 \( 1 - 0.985T + 89T^{2} \)
97 \( 1 - 6.97T + 97T^{2} \)
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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

−7.88316668019802460937563267163, −7.48610207608439437210348637527, −6.91589313524561851432897011590, −6.24311521380666917764443186481, −5.10974374612075417165286981729, −4.06188454232302426450475522434, −3.26643344050205215367148008230, −2.67933325067097264924793161149, −1.13110758672522219657796484932, −0.62310000263061230541292800944, 0.62310000263061230541292800944, 1.13110758672522219657796484932, 2.67933325067097264924793161149, 3.26643344050205215367148008230, 4.06188454232302426450475522434, 5.10974374612075417165286981729, 6.24311521380666917764443186481, 6.91589313524561851432897011590, 7.48610207608439437210348637527, 7.88316668019802460937563267163

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