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  − 2.08·2-s + 2.33·4-s − 1.33·5-s + 7-s − 0.698·8-s + 2.77·10-s + 5.74·13-s − 2.08·14-s − 3.21·16-s − 4.27·17-s + 0.162·19-s − 3.11·20-s − 1.12·23-s − 3.22·25-s − 11.9·26-s + 2.33·28-s − 9.83·29-s + 9.09·31-s + 8.09·32-s + 8.89·34-s − 1.33·35-s − 8.42·37-s − 0.338·38-s + 0.929·40-s − 4.50·41-s − 8.02·43-s + 2.34·46-s + ⋯
L(s)  = 1  − 1.47·2-s + 1.16·4-s − 0.595·5-s + 0.377·7-s − 0.246·8-s + 0.877·10-s + 1.59·13-s − 0.556·14-s − 0.804·16-s − 1.03·17-s + 0.0372·19-s − 0.695·20-s − 0.234·23-s − 0.645·25-s − 2.34·26-s + 0.441·28-s − 1.82·29-s + 1.63·31-s + 1.43·32-s + 1.52·34-s − 0.225·35-s − 1.38·37-s − 0.0549·38-s + 0.147·40-s − 0.703·41-s − 1.22·43-s + 0.345·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6773343283$
$L(\frac12)$  $\approx$  $0.6773343283$
$L(\frac{3}{2})$   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 \( 1 + 2.08T + 2T^{2} \)
5 \( 1 + 1.33T + 5T^{2} \)
13 \( 1 - 5.74T + 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
19 \( 1 - 0.162T + 19T^{2} \)
23 \( 1 + 1.12T + 23T^{2} \)
29 \( 1 + 9.83T + 29T^{2} \)
31 \( 1 - 9.09T + 31T^{2} \)
37 \( 1 + 8.42T + 37T^{2} \)
41 \( 1 + 4.50T + 41T^{2} \)
43 \( 1 + 8.02T + 43T^{2} \)
47 \( 1 - 2.79T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 - 8.06T + 61T^{2} \)
67 \( 1 - 9.10T + 67T^{2} \)
71 \( 1 + 1.01T + 71T^{2} \)
73 \( 1 + 7.64T + 73T^{2} \)
79 \( 1 + 0.00547T + 79T^{2} \)
83 \( 1 + 2.83T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−8.208469573948024816032726636171, −7.30758919058795372706315327256, −6.83321268058255774567995204875, −6.02936513313512584937815678618, −5.11743772136726211808109922224, −4.10708263122627574965094054010, −3.60432105934473122457382297798, −2.26697792798948959552367046762, −1.55916566296898479130093082628, −0.53473144094150101478003195739, 0.53473144094150101478003195739, 1.55916566296898479130093082628, 2.26697792798948959552367046762, 3.60432105934473122457382297798, 4.10708263122627574965094054010, 5.11743772136726211808109922224, 6.02936513313512584937815678618, 6.83321268058255774567995204875, 7.30758919058795372706315327256, 8.208469573948024816032726636171

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