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  + 1.46·2-s + 0.139·4-s − 2.39·5-s + 7-s − 2.72·8-s − 3.50·10-s − 5.04·13-s + 1.46·14-s − 4.25·16-s − 6.36·17-s + 5.32·19-s − 0.333·20-s − 4.92·23-s + 0.751·25-s − 7.37·26-s + 0.139·28-s + 5.04·29-s − 7.57·31-s − 0.786·32-s − 9.31·34-s − 2.39·35-s + 4.24·37-s + 7.78·38-s + 6.52·40-s − 0.646·41-s + 10.5·43-s − 7.20·46-s + ⋯
L(s)  = 1  + 1.03·2-s + 0.0695·4-s − 1.07·5-s + 0.377·7-s − 0.962·8-s − 1.10·10-s − 1.39·13-s + 0.390·14-s − 1.06·16-s − 1.54·17-s + 1.22·19-s − 0.0746·20-s − 1.02·23-s + 0.150·25-s − 1.44·26-s + 0.0263·28-s + 0.936·29-s − 1.35·31-s − 0.138·32-s − 1.59·34-s − 0.405·35-s + 0.698·37-s + 1.26·38-s + 1.03·40-s − 0.101·41-s + 1.60·43-s − 1.06·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.322058887$
$L(\frac12)$  $\approx$  $1.322058887$
$L(\frac{3}{2})$   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 \( 1 - 1.46T + 2T^{2} \)
5 \( 1 + 2.39T + 5T^{2} \)
13 \( 1 + 5.04T + 13T^{2} \)
17 \( 1 + 6.36T + 17T^{2} \)
19 \( 1 - 5.32T + 19T^{2} \)
23 \( 1 + 4.92T + 23T^{2} \)
29 \( 1 - 5.04T + 29T^{2} \)
31 \( 1 + 7.57T + 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + 0.646T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 0.526T + 47T^{2} \)
53 \( 1 + 3.72T + 53T^{2} \)
59 \( 1 + 7.97T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 8.76T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 1.87T + 97T^{2} \)
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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

−7.79461256769360828674849544806, −7.16074104159883259390767368823, −6.43324305533865202982805985774, −5.51872664774747092720448766797, −4.91779128438094852679397298397, −4.29525487019604110628010507227, −3.82648759130211974008494397741, −2.87278688441794125276318957341, −2.13709266336475789557991943183, −0.46494875627773215504581136371, 0.46494875627773215504581136371, 2.13709266336475789557991943183, 2.87278688441794125276318957341, 3.82648759130211974008494397741, 4.29525487019604110628010507227, 4.91779128438094852679397298397, 5.51872664774747092720448766797, 6.43324305533865202982805985774, 7.16074104159883259390767368823, 7.79461256769360828674849544806

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