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  + 2.63·2-s + 4.96·4-s − 3.08·5-s + 7-s + 7.83·8-s − 8.14·10-s + 5.64·13-s + 2.63·14-s + 10.7·16-s − 5.60·17-s − 0.122·19-s − 15.3·20-s − 1.67·23-s + 4.50·25-s + 14.8·26-s + 4.96·28-s + 8.85·29-s + 1.59·31-s + 12.7·32-s − 14.7·34-s − 3.08·35-s + 4.17·37-s − 0.323·38-s − 24.1·40-s + 3.48·41-s + 5.10·43-s − 4.41·46-s + ⋯
L(s)  = 1  + 1.86·2-s + 2.48·4-s − 1.37·5-s + 0.377·7-s + 2.77·8-s − 2.57·10-s + 1.56·13-s + 0.705·14-s + 2.68·16-s − 1.35·17-s − 0.0281·19-s − 3.42·20-s − 0.348·23-s + 0.901·25-s + 2.92·26-s + 0.939·28-s + 1.64·29-s + 0.286·31-s + 2.24·32-s − 2.53·34-s − 0.521·35-s + 0.685·37-s − 0.0525·38-s − 3.82·40-s + 0.544·41-s + 0.778·43-s − 0.651·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.327608478$
$L(\frac12)$  $\approx$  $6.327608478$
$L(\frac{3}{2})$   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 \( 1 - 2.63T + 2T^{2} \)
5 \( 1 + 3.08T + 5T^{2} \)
13 \( 1 - 5.64T + 13T^{2} \)
17 \( 1 + 5.60T + 17T^{2} \)
19 \( 1 + 0.122T + 19T^{2} \)
23 \( 1 + 1.67T + 23T^{2} \)
29 \( 1 - 8.85T + 29T^{2} \)
31 \( 1 - 1.59T + 31T^{2} \)
37 \( 1 - 4.17T + 37T^{2} \)
41 \( 1 - 3.48T + 41T^{2} \)
43 \( 1 - 5.10T + 43T^{2} \)
47 \( 1 - 1.59T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + 6.61T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 + 8.04T + 67T^{2} \)
71 \( 1 + 6.24T + 71T^{2} \)
73 \( 1 + 3.51T + 73T^{2} \)
79 \( 1 - 9.39T + 79T^{2} \)
83 \( 1 + 9.43T + 83T^{2} \)
89 \( 1 - 8.45T + 89T^{2} \)
97 \( 1 + 5.68T + 97T^{2} \)
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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

−7.63086071667691367634930475287, −6.97969131943926136207431927571, −6.28968285776067312994142002213, −5.78243208761734466363761945245, −4.68944832703796891598124598055, −4.32047056501552366532608717961, −3.81615072551496141417285749692, −3.03429395249518844120204868665, −2.20974488078276763235920640659, −0.976339835528902820581426339494, 0.976339835528902820581426339494, 2.20974488078276763235920640659, 3.03429395249518844120204868665, 3.81615072551496141417285749692, 4.32047056501552366532608717961, 4.68944832703796891598124598055, 5.78243208761734466363761945245, 6.28968285776067312994142002213, 6.97969131943926136207431927571, 7.63086071667691367634930475287

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