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  + 0.899·2-s − 1.19·4-s − 0.803·5-s + 7-s − 2.87·8-s − 0.723·10-s + 2.94·13-s + 0.899·14-s − 0.202·16-s + 3.30·17-s + 7.93·19-s + 0.956·20-s − 7.53·23-s − 4.35·25-s + 2.65·26-s − 1.19·28-s − 0.234·29-s + 3.98·31-s + 5.55·32-s + 2.97·34-s − 0.803·35-s − 0.536·37-s + 7.14·38-s + 2.30·40-s − 4.78·41-s + 3.16·43-s − 6.78·46-s + ⋯
L(s)  = 1  + 0.636·2-s − 0.595·4-s − 0.359·5-s + 0.377·7-s − 1.01·8-s − 0.228·10-s + 0.817·13-s + 0.240·14-s − 0.0506·16-s + 0.801·17-s + 1.82·19-s + 0.213·20-s − 1.57·23-s − 0.870·25-s + 0.520·26-s − 0.224·28-s − 0.0435·29-s + 0.715·31-s + 0.982·32-s + 0.509·34-s − 0.135·35-s − 0.0881·37-s + 1.15·38-s + 0.364·40-s − 0.747·41-s + 0.482·43-s − 0.999·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.137514916$
$L(\frac12)$  $\approx$  $2.137514916$
$L(\frac{3}{2})$   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 \( 1 - 0.899T + 2T^{2} \)
5 \( 1 + 0.803T + 5T^{2} \)
13 \( 1 - 2.94T + 13T^{2} \)
17 \( 1 - 3.30T + 17T^{2} \)
19 \( 1 - 7.93T + 19T^{2} \)
23 \( 1 + 7.53T + 23T^{2} \)
29 \( 1 + 0.234T + 29T^{2} \)
31 \( 1 - 3.98T + 31T^{2} \)
37 \( 1 + 0.536T + 37T^{2} \)
41 \( 1 + 4.78T + 41T^{2} \)
43 \( 1 - 3.16T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + 5.25T + 59T^{2} \)
61 \( 1 - 3.65T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 8.90T + 73T^{2} \)
79 \( 1 + 8.37T + 79T^{2} \)
83 \( 1 + 7.26T + 83T^{2} \)
89 \( 1 - 8.03T + 89T^{2} \)
97 \( 1 - 19.2T + 97T^{2} \)
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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

−7.88758714900953202865035893140, −7.31885272671763594925029777394, −6.12535689930317450294953953992, −5.77820056899992430371686003570, −5.01156410148587779162232121448, −4.30428622727260892762362083611, −3.57100340654412053659586054484, −3.10382264443307061835366118769, −1.75517666166593304690105135295, −0.68522746987687319911709884697, 0.68522746987687319911709884697, 1.75517666166593304690105135295, 3.10382264443307061835366118769, 3.57100340654412053659586054484, 4.30428622727260892762362083611, 5.01156410148587779162232121448, 5.77820056899992430371686003570, 6.12535689930317450294953953992, 7.31885272671763594925029777394, 7.88758714900953202865035893140

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