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  + 2.10·2-s + 2.42·4-s + 3.75·5-s + 7-s + 0.897·8-s + 7.89·10-s + 3.46·13-s + 2.10·14-s − 2.96·16-s − 1.52·17-s − 5.16·19-s + 9.10·20-s + 4.87·23-s + 9.06·25-s + 7.29·26-s + 2.42·28-s − 10.7·29-s + 8.12·31-s − 8.03·32-s − 3.21·34-s + 3.75·35-s + 9.25·37-s − 10.8·38-s + 3.36·40-s + 10.8·41-s − 0.137·43-s + 10.2·46-s + ⋯
L(s)  = 1  + 1.48·2-s + 1.21·4-s + 1.67·5-s + 0.377·7-s + 0.317·8-s + 2.49·10-s + 0.961·13-s + 0.562·14-s − 0.741·16-s − 0.370·17-s − 1.18·19-s + 2.03·20-s + 1.01·23-s + 1.81·25-s + 1.43·26-s + 0.458·28-s − 1.99·29-s + 1.45·31-s − 1.42·32-s − 0.550·34-s + 0.633·35-s + 1.52·37-s − 1.76·38-s + 0.532·40-s + 1.70·41-s − 0.0209·43-s + 1.51·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.246928445$
$L(\frac12)$  $\approx$  $7.246928445$
$L(\frac{3}{2})$   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 \( 1 - 2.10T + 2T^{2} \)
5 \( 1 - 3.75T + 5T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + 1.52T + 17T^{2} \)
19 \( 1 + 5.16T + 19T^{2} \)
23 \( 1 - 4.87T + 23T^{2} \)
29 \( 1 + 10.7T + 29T^{2} \)
31 \( 1 - 8.12T + 31T^{2} \)
37 \( 1 - 9.25T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 0.137T + 43T^{2} \)
47 \( 1 - 2.73T + 47T^{2} \)
53 \( 1 - 2.79T + 53T^{2} \)
59 \( 1 - 4.84T + 59T^{2} \)
61 \( 1 + 0.297T + 61T^{2} \)
67 \( 1 - 0.816T + 67T^{2} \)
71 \( 1 + 5.72T + 71T^{2} \)
73 \( 1 + 2.47T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 + 8.91T + 89T^{2} \)
97 \( 1 - 8.08T + 97T^{2} \)
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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

−7.66865847722296229543423793129, −6.72255249933448121822071404928, −6.13507669650928628613260268486, −5.84764147004327925743432882678, −5.08292142485640796619824689717, −4.41760008061637534516346444046, −3.71003283419183877759267340019, −2.59476383956658201454084308411, −2.22330573437325834757026697826, −1.15242388409361635691698467271, 1.15242388409361635691698467271, 2.22330573437325834757026697826, 2.59476383956658201454084308411, 3.71003283419183877759267340019, 4.41760008061637534516346444046, 5.08292142485640796619824689717, 5.84764147004327925743432882678, 6.13507669650928628613260268486, 6.72255249933448121822071404928, 7.66865847722296229543423793129

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