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  − 2.76·2-s + 5.62·4-s + 2.62·5-s + 7-s − 10.0·8-s − 7.25·10-s − 2.38·13-s − 2.76·14-s + 16.4·16-s + 2.38·17-s − 1.72·19-s + 14.7·20-s + 0.626·23-s + 1.89·25-s + 6.59·26-s + 5.62·28-s − 1.72·29-s − 2.23·31-s − 25.2·32-s − 6.59·34-s + 2.62·35-s − 6.89·37-s + 4.77·38-s − 26.2·40-s − 10.3·41-s + 7.25·43-s − 1.72·46-s + ⋯
L(s)  = 1  − 1.95·2-s + 2.81·4-s + 1.17·5-s + 0.377·7-s − 3.54·8-s − 2.29·10-s − 0.662·13-s − 0.738·14-s + 4.10·16-s + 0.579·17-s − 0.396·19-s + 3.30·20-s + 0.130·23-s + 0.379·25-s + 1.29·26-s + 1.06·28-s − 0.321·29-s − 0.402·31-s − 4.46·32-s − 1.13·34-s + 0.443·35-s − 1.13·37-s + 0.774·38-s − 4.15·40-s − 1.62·41-s + 1.10·43-s − 0.254·46-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 \( 1 + 2.76T + 2T^{2} \)
5 \( 1 - 2.62T + 5T^{2} \)
13 \( 1 + 2.38T + 13T^{2} \)
17 \( 1 - 2.38T + 17T^{2} \)
19 \( 1 + 1.72T + 19T^{2} \)
23 \( 1 - 0.626T + 23T^{2} \)
29 \( 1 + 1.72T + 29T^{2} \)
31 \( 1 + 2.23T + 31T^{2} \)
37 \( 1 + 6.89T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 7.25T + 43T^{2} \)
47 \( 1 + 6.38T + 47T^{2} \)
53 \( 1 - 9.25T + 53T^{2} \)
59 \( 1 - 1.76T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 6.42T + 67T^{2} \)
71 \( 1 + 8.08T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 - 15.2T + 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + 8.35T + 97T^{2} \)
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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

−7.64509336779221471897249915314, −7.06711674107894364964828474146, −6.39630004025078393758070455087, −5.71460385171020972356921982996, −5.05047250523704989371899821261, −3.53382557456664855930680214009, −2.56828221974674493143531529110, −1.92857691161388760442532291467, −1.28509090457716517419825559442, 0, 1.28509090457716517419825559442, 1.92857691161388760442532291467, 2.56828221974674493143531529110, 3.53382557456664855930680214009, 5.05047250523704989371899821261, 5.71460385171020972356921982996, 6.39630004025078393758070455087, 7.06711674107894364964828474146, 7.64509336779221471897249915314

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