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  − 0.790·2-s − 1.37·4-s + 4.15·5-s − 7-s + 2.66·8-s − 3.28·10-s − 2.46·13-s + 0.790·14-s + 0.639·16-s − 1.99·17-s − 3.11·19-s − 5.70·20-s − 3.45·23-s + 12.2·25-s + 1.95·26-s + 1.37·28-s − 3.32·29-s − 0.288·31-s − 5.84·32-s + 1.57·34-s − 4.15·35-s + 7.59·37-s + 2.46·38-s + 11.0·40-s + 2.32·41-s + 8.35·43-s + 2.73·46-s + ⋯
L(s)  = 1  − 0.559·2-s − 0.687·4-s + 1.85·5-s − 0.377·7-s + 0.943·8-s − 1.03·10-s − 0.684·13-s + 0.211·14-s + 0.159·16-s − 0.483·17-s − 0.713·19-s − 1.27·20-s − 0.721·23-s + 2.44·25-s + 0.382·26-s + 0.259·28-s − 0.617·29-s − 0.0518·31-s − 1.03·32-s + 0.270·34-s − 0.701·35-s + 1.24·37-s + 0.399·38-s + 1.75·40-s + 0.363·41-s + 1.27·43-s + 0.403·46-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 \( 1 + 0.790T + 2T^{2} \)
5 \( 1 - 4.15T + 5T^{2} \)
13 \( 1 + 2.46T + 13T^{2} \)
17 \( 1 + 1.99T + 17T^{2} \)
19 \( 1 + 3.11T + 19T^{2} \)
23 \( 1 + 3.45T + 23T^{2} \)
29 \( 1 + 3.32T + 29T^{2} \)
31 \( 1 + 0.288T + 31T^{2} \)
37 \( 1 - 7.59T + 37T^{2} \)
41 \( 1 - 2.32T + 41T^{2} \)
43 \( 1 - 8.35T + 43T^{2} \)
47 \( 1 + 2.94T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 - 7.82T + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 + 8.39T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 8.86T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 + 4.58T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 7.28T + 97T^{2} \)
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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

−7.64491755081687281319767804335, −6.75301198797533318894046549717, −6.10603971014902218096944556256, −5.54615383688021959336119001853, −4.75062579112254034027997370529, −4.10089176788230212561033638677, −2.82409030659416935175810715215, −2.11359987935115270506727766361, −1.30129153509314971184286979385, 0, 1.30129153509314971184286979385, 2.11359987935115270506727766361, 2.82409030659416935175810715215, 4.10089176788230212561033638677, 4.75062579112254034027997370529, 5.54615383688021959336119001853, 6.10603971014902218096944556256, 6.75301198797533318894046549717, 7.64491755081687281319767804335

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