Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.11·2-s − 0.757·4-s − 3.45·5-s + 7-s + 3.07·8-s + 3.85·10-s − 2.05·13-s − 1.11·14-s − 1.90·16-s − 1.93·17-s + 1.62·19-s + 2.61·20-s + 0.807·23-s + 6.94·25-s + 2.29·26-s − 0.757·28-s − 7.97·29-s + 0.788·31-s − 4.01·32-s + 2.15·34-s − 3.45·35-s + 10.0·37-s − 1.80·38-s − 10.6·40-s + 2.12·41-s + 3.08·43-s − 0.899·46-s + ⋯
L(s)  = 1  − 0.788·2-s − 0.378·4-s − 1.54·5-s + 0.377·7-s + 1.08·8-s + 1.21·10-s − 0.571·13-s − 0.297·14-s − 0.477·16-s − 0.468·17-s + 0.372·19-s + 0.585·20-s + 0.168·23-s + 1.38·25-s + 0.450·26-s − 0.143·28-s − 1.48·29-s + 0.141·31-s − 0.710·32-s + 0.369·34-s − 0.584·35-s + 1.65·37-s − 0.293·38-s − 1.67·40-s + 0.332·41-s + 0.469·43-s − 0.132·46-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  =  1
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 1.11T + 2T^{2} \)
5 \( 1 + 3.45T + 5T^{2} \)
13 \( 1 + 2.05T + 13T^{2} \)
17 \( 1 + 1.93T + 17T^{2} \)
19 \( 1 - 1.62T + 19T^{2} \)
23 \( 1 - 0.807T + 23T^{2} \)
29 \( 1 + 7.97T + 29T^{2} \)
31 \( 1 - 0.788T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 - 2.12T + 41T^{2} \)
43 \( 1 - 3.08T + 43T^{2} \)
47 \( 1 + 7.56T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 3.29T + 59T^{2} \)
61 \( 1 + 1.07T + 61T^{2} \)
67 \( 1 - 2.40T + 67T^{2} \)
71 \( 1 - 3.18T + 71T^{2} \)
73 \( 1 - 1.22T + 73T^{2} \)
79 \( 1 + 9.48T + 79T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 - 4.43T + 89T^{2} \)
97 \( 1 - 6.46T + 97T^{2} \)
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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

−7.69640020050748799146876615266, −7.30077377649955457703628152762, −6.32131052612072838402247834987, −5.18782486943316062469264150170, −4.59728380722246166610010939480, −4.02475171414194152353661034346, −3.24223664499633320035966494965, −2.05881944265911279756266484057, −0.888494675307936829097359807989, 0, 0.888494675307936829097359807989, 2.05881944265911279756266484057, 3.24223664499633320035966494965, 4.02475171414194152353661034346, 4.59728380722246166610010939480, 5.18782486943316062469264150170, 6.32131052612072838402247834987, 7.30077377649955457703628152762, 7.69640020050748799146876615266

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