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  + 0.400·2-s − 1.83·4-s + 2.30·5-s − 7-s − 1.53·8-s + 0.924·10-s − 4.49·13-s − 0.400·14-s + 3.06·16-s − 7.46·17-s + 1.34·19-s − 4.24·20-s − 4.27·23-s + 0.315·25-s − 1.80·26-s + 1.83·28-s − 1.12·29-s − 8.11·31-s + 4.30·32-s − 2.99·34-s − 2.30·35-s + 11.0·37-s + 0.540·38-s − 3.54·40-s + 1.16·41-s − 9.95·43-s − 1.71·46-s + ⋯
L(s)  = 1  + 0.283·2-s − 0.919·4-s + 1.03·5-s − 0.377·7-s − 0.544·8-s + 0.292·10-s − 1.24·13-s − 0.107·14-s + 0.765·16-s − 1.81·17-s + 0.309·19-s − 0.948·20-s − 0.890·23-s + 0.0631·25-s − 0.353·26-s + 0.347·28-s − 0.209·29-s − 1.45·31-s + 0.761·32-s − 0.513·34-s − 0.389·35-s + 1.81·37-s + 0.0876·38-s − 0.561·40-s + 0.181·41-s − 1.51·43-s − 0.252·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.287651475$
$L(\frac12)$  $\approx$  $1.287651475$
$L(\frac{3}{2})$   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 \( 1 - 0.400T + 2T^{2} \)
5 \( 1 - 2.30T + 5T^{2} \)
13 \( 1 + 4.49T + 13T^{2} \)
17 \( 1 + 7.46T + 17T^{2} \)
19 \( 1 - 1.34T + 19T^{2} \)
23 \( 1 + 4.27T + 23T^{2} \)
29 \( 1 + 1.12T + 29T^{2} \)
31 \( 1 + 8.11T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 - 1.16T + 41T^{2} \)
43 \( 1 + 9.95T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + 1.04T + 59T^{2} \)
61 \( 1 - 6.69T + 61T^{2} \)
67 \( 1 + 2.46T + 67T^{2} \)
71 \( 1 - 8.42T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + 2.71T + 79T^{2} \)
83 \( 1 - 9.12T + 83T^{2} \)
89 \( 1 - 5.28T + 89T^{2} \)
97 \( 1 - 5.74T + 97T^{2} \)
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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

−7.87816399042136076642005814509, −7.09912293940124523818083002558, −6.34493740726714836373454757847, −5.67988249389399473456645668651, −5.12650061287962381890699217226, −4.33897455033229839076844120531, −3.72009598520973932495753047671, −2.52506074127189937462144113892, −2.04998694651984969781361761662, −0.51998562932322066522382746385, 0.51998562932322066522382746385, 2.04998694651984969781361761662, 2.52506074127189937462144113892, 3.72009598520973932495753047671, 4.33897455033229839076844120531, 5.12650061287962381890699217226, 5.67988249389399473456645668651, 6.34493740726714836373454757847, 7.09912293940124523818083002558, 7.87816399042136076642005814509

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