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  − 1.93·2-s + 1.74·4-s − 4.18·5-s + 7-s + 0.491·8-s + 8.10·10-s + 3.17·13-s − 1.93·14-s − 4.44·16-s + 6.85·17-s + 0.318·19-s − 7.31·20-s + 1.87·23-s + 12.5·25-s − 6.14·26-s + 1.74·28-s − 3.17·29-s + 9.23·31-s + 7.61·32-s − 13.2·34-s − 4.18·35-s − 7.55·37-s − 0.616·38-s − 2.06·40-s + 9.36·41-s + 10.8·43-s − 3.62·46-s + ⋯
L(s)  = 1  − 1.36·2-s + 0.872·4-s − 1.87·5-s + 0.377·7-s + 0.173·8-s + 2.56·10-s + 0.880·13-s − 0.517·14-s − 1.11·16-s + 1.66·17-s + 0.0731·19-s − 1.63·20-s + 0.390·23-s + 2.51·25-s − 1.20·26-s + 0.329·28-s − 0.589·29-s + 1.65·31-s + 1.34·32-s − 2.27·34-s − 0.708·35-s − 1.24·37-s − 0.100·38-s − 0.325·40-s + 1.46·41-s + 1.66·43-s − 0.533·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8235119343$
$L(\frac12)$  $\approx$  $0.8235119343$
$L(\frac{3}{2})$   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 \( 1 + 1.93T + 2T^{2} \)
5 \( 1 + 4.18T + 5T^{2} \)
13 \( 1 - 3.17T + 13T^{2} \)
17 \( 1 - 6.85T + 17T^{2} \)
19 \( 1 - 0.318T + 19T^{2} \)
23 \( 1 - 1.87T + 23T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 - 9.23T + 31T^{2} \)
37 \( 1 + 7.55T + 37T^{2} \)
41 \( 1 - 9.36T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 - 8.06T + 47T^{2} \)
53 \( 1 + 0.508T + 53T^{2} \)
59 \( 1 - 7.04T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 2.66T + 67T^{2} \)
71 \( 1 - 5.01T + 71T^{2} \)
73 \( 1 - 4.82T + 73T^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 - 3.52T + 83T^{2} \)
89 \( 1 - 1.74T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−7.88108623830344832428964471452, −7.53793777864237757141100445931, −6.98573868749922407516357384770, −5.93707222749737771597083316821, −4.96084980957516028206383358788, −4.14349500596843577759537652272, −3.59112960616472198428679165817, −2.60968418637354136517134876760, −1.15775519602384426029167464805, −0.70461931020039156931083714419, 0.70461931020039156931083714419, 1.15775519602384426029167464805, 2.60968418637354136517134876760, 3.59112960616472198428679165817, 4.14349500596843577759537652272, 4.96084980957516028206383358788, 5.93707222749737771597083316821, 6.98573868749922407516357384770, 7.53793777864237757141100445931, 7.88108623830344832428964471452

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