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  + 0.301·2-s − 1.90·4-s + 0.698·5-s + 7-s − 1.17·8-s + 0.210·10-s − 4.43·13-s + 0.301·14-s + 3.46·16-s − 6.80·17-s − 1.78·19-s − 1.33·20-s + 9.20·23-s − 4.51·25-s − 1.33·26-s − 1.90·28-s + 7.89·29-s + 10.4·31-s + 3.39·32-s − 2.04·34-s + 0.698·35-s + 6.16·37-s − 0.538·38-s − 0.822·40-s + 1.33·41-s − 6.33·43-s + 2.77·46-s + ⋯
L(s)  = 1  + 0.212·2-s − 0.954·4-s + 0.312·5-s + 0.377·7-s − 0.416·8-s + 0.0665·10-s − 1.22·13-s + 0.0804·14-s + 0.866·16-s − 1.65·17-s − 0.410·19-s − 0.298·20-s + 1.91·23-s − 0.902·25-s − 0.261·26-s − 0.360·28-s + 1.46·29-s + 1.87·31-s + 0.600·32-s − 0.351·34-s + 0.118·35-s + 1.01·37-s − 0.0874·38-s − 0.130·40-s + 0.208·41-s − 0.965·43-s + 0.408·46-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 \( 1 - 0.301T + 2T^{2} \)
5 \( 1 - 0.698T + 5T^{2} \)
13 \( 1 + 4.43T + 13T^{2} \)
17 \( 1 + 6.80T + 17T^{2} \)
19 \( 1 + 1.78T + 19T^{2} \)
23 \( 1 - 9.20T + 23T^{2} \)
29 \( 1 - 7.89T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 - 6.16T + 37T^{2} \)
41 \( 1 - 1.33T + 41T^{2} \)
43 \( 1 + 6.33T + 43T^{2} \)
47 \( 1 + 8.20T + 47T^{2} \)
53 \( 1 - 2.35T + 53T^{2} \)
59 \( 1 - 4.76T + 59T^{2} \)
61 \( 1 - 1.69T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + 7.58T + 71T^{2} \)
73 \( 1 + 6.96T + 73T^{2} \)
79 \( 1 + 8.85T + 79T^{2} \)
83 \( 1 - 5.71T + 83T^{2} \)
89 \( 1 - 2.22T + 89T^{2} \)
97 \( 1 + 0.171T + 97T^{2} \)
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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

−7.57089658697639481994708473944, −6.72048755238875825824261180701, −6.17883981924136073356321494190, −5.14118472422754750640558160063, −4.66008900148433388667818920308, −4.29917916464032888887278041384, −3.00399513216261021161541558313, −2.43282157038605634055200820796, −1.17288776228353016214387945567, 0, 1.17288776228353016214387945567, 2.43282157038605634055200820796, 3.00399513216261021161541558313, 4.29917916464032888887278041384, 4.66008900148433388667818920308, 5.14118472422754750640558160063, 6.17883981924136073356321494190, 6.72048755238875825824261180701, 7.57089658697639481994708473944

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