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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $10.23236668$
$L(\frac12)$  $\approx$  $10.23236668$
$L(\frac{3}{2})$   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

−7.51900503828649394355083112699, −6.64844196405367210642320020009, −6.07314018483608838530808611512, −5.96553406263677730007712425683, −5.07486280643750122434109152699, −4.50776201765889904366122806142, −3.49370637109205258610071642854, −2.85181636006035773028079371244, −2.07435248470901871098373589998, −1.40661473689758627434928155529, 1.40661473689758627434928155529, 2.07435248470901871098373589998, 2.85181636006035773028079371244, 3.49370637109205258610071642854, 4.50776201765889904366122806142, 5.07486280643750122434109152699, 5.96553406263677730007712425683, 6.07314018483608838530808611512, 6.64844196405367210642320020009, 7.51900503828649394355083112699

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