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  − 1.33·2-s − 0.222·4-s + 0.873·5-s + 7-s + 2.96·8-s − 1.16·10-s − 0.0395·13-s − 1.33·14-s − 3.50·16-s + 2.04·17-s + 5.69·19-s − 0.194·20-s − 5.97·23-s − 4.23·25-s + 0.0527·26-s − 0.222·28-s − 4.67·29-s + 7.47·31-s − 1.25·32-s − 2.72·34-s + 0.873·35-s + 11.5·37-s − 7.59·38-s + 2.58·40-s − 6.14·41-s + 1.79·43-s + 7.97·46-s + ⋯
L(s)  = 1  − 0.942·2-s − 0.111·4-s + 0.390·5-s + 0.377·7-s + 1.04·8-s − 0.368·10-s − 0.0109·13-s − 0.356·14-s − 0.876·16-s + 0.496·17-s + 1.30·19-s − 0.0434·20-s − 1.24·23-s − 0.847·25-s + 0.0103·26-s − 0.0420·28-s − 0.867·29-s + 1.34·31-s − 0.221·32-s − 0.467·34-s + 0.147·35-s + 1.90·37-s − 1.23·38-s + 0.409·40-s − 0.960·41-s + 0.274·43-s + 1.17·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.222364687$
$L(\frac12)$  $\approx$  $1.222364687$
$L(\frac{3}{2})$   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 \( 1 + 1.33T + 2T^{2} \)
5 \( 1 - 0.873T + 5T^{2} \)
13 \( 1 + 0.0395T + 13T^{2} \)
17 \( 1 - 2.04T + 17T^{2} \)
19 \( 1 - 5.69T + 19T^{2} \)
23 \( 1 + 5.97T + 23T^{2} \)
29 \( 1 + 4.67T + 29T^{2} \)
31 \( 1 - 7.47T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 + 6.14T + 41T^{2} \)
43 \( 1 - 1.79T + 43T^{2} \)
47 \( 1 + 6.04T + 47T^{2} \)
53 \( 1 - 0.124T + 53T^{2} \)
59 \( 1 - 14.3T + 59T^{2} \)
61 \( 1 + 7.55T + 61T^{2} \)
67 \( 1 - 4.85T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + 0.612T + 73T^{2} \)
79 \( 1 - 0.155T + 79T^{2} \)
83 \( 1 - 8.91T + 83T^{2} \)
89 \( 1 + 5.17T + 89T^{2} \)
97 \( 1 - 7.94T + 97T^{2} \)
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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

−7.970392396808035078880456777831, −7.52929335646350145472068569200, −6.60262896155004790208719335802, −5.76751805850203957376105178856, −5.14733116046636859512971537620, −4.32260408509161302825666487724, −3.56413763635824494282096846282, −2.40156481697510883925838139051, −1.56317046483898476247329158466, −0.68092491107130066562941294686, 0.68092491107130066562941294686, 1.56317046483898476247329158466, 2.40156481697510883925838139051, 3.56413763635824494282096846282, 4.32260408509161302825666487724, 5.14733116046636859512971537620, 5.76751805850203957376105178856, 6.60262896155004790208719335802, 7.52929335646350145472068569200, 7.970392396808035078880456777831

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