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  + 2.61·2-s + 4.85·4-s − 5-s + 7-s + 7.47·8-s − 2.61·10-s + 3.23·13-s + 2.61·14-s + 9.85·16-s + 8.09·17-s − 6.23·19-s − 4.85·20-s + 6.09·23-s − 4·25-s + 8.47·26-s + 4.85·28-s + 2.38·29-s + 0.236·31-s + 10.8·32-s + 21.1·34-s − 35-s − 2.47·37-s − 16.3·38-s − 7.47·40-s − 11.1·41-s + 7.56·43-s + 15.9·46-s + ⋯
L(s)  = 1  + 1.85·2-s + 2.42·4-s − 0.447·5-s + 0.377·7-s + 2.64·8-s − 0.827·10-s + 0.897·13-s + 0.699·14-s + 2.46·16-s + 1.96·17-s − 1.43·19-s − 1.08·20-s + 1.26·23-s − 0.800·25-s + 1.66·26-s + 0.917·28-s + 0.442·29-s + 0.0423·31-s + 1.91·32-s + 3.63·34-s − 0.169·35-s − 0.406·37-s − 2.64·38-s − 1.18·40-s − 1.74·41-s + 1.15·43-s + 2.35·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.698838552$
$L(\frac12)$  $\approx$  $7.698838552$
$L(\frac{3}{2})$   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 \( 1 - 2.61T + 2T^{2} \)
5 \( 1 + T + 5T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
17 \( 1 - 8.09T + 17T^{2} \)
19 \( 1 + 6.23T + 19T^{2} \)
23 \( 1 - 6.09T + 23T^{2} \)
29 \( 1 - 2.38T + 29T^{2} \)
31 \( 1 - 0.236T + 31T^{2} \)
37 \( 1 + 2.47T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 - 7.56T + 43T^{2} \)
47 \( 1 - 4.38T + 47T^{2} \)
53 \( 1 - 4.61T + 53T^{2} \)
59 \( 1 - 0.0901T + 59T^{2} \)
61 \( 1 + 5.38T + 61T^{2} \)
67 \( 1 - 7.32T + 67T^{2} \)
71 \( 1 - 4.90T + 71T^{2} \)
73 \( 1 + 9.76T + 73T^{2} \)
79 \( 1 - 8.61T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + 0.145T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−7.68934159419656972091396200603, −6.92586723228949481823625754002, −6.29298224933793876586478225372, −5.58997789516078427970213017297, −5.07185891220514493673195283124, −4.24324501819282769733137107586, −3.66572637147918686773064194373, −3.08612920668107643081192969018, −2.09908141979025997299888366865, −1.13051217297041408005331222061, 1.13051217297041408005331222061, 2.09908141979025997299888366865, 3.08612920668107643081192969018, 3.66572637147918686773064194373, 4.24324501819282769733137107586, 5.07185891220514493673195283124, 5.58997789516078427970213017297, 6.29298224933793876586478225372, 6.92586723228949481823625754002, 7.68934159419656972091396200603

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