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  + 0.823·2-s − 1.32·4-s − 2.98·5-s + 7-s − 2.73·8-s − 2.45·10-s + 2.20·13-s + 0.823·14-s + 0.390·16-s − 4.40·17-s − 1.72·19-s + 3.94·20-s + 8.39·23-s + 3.91·25-s + 1.81·26-s − 1.32·28-s − 3.29·29-s + 7.47·31-s + 5.79·32-s − 3.62·34-s − 2.98·35-s − 8.78·37-s − 1.42·38-s + 8.16·40-s − 5.39·41-s + 9.44·43-s + 6.91·46-s + ⋯
L(s)  = 1  + 0.582·2-s − 0.660·4-s − 1.33·5-s + 0.377·7-s − 0.967·8-s − 0.777·10-s + 0.610·13-s + 0.220·14-s + 0.0976·16-s − 1.06·17-s − 0.395·19-s + 0.882·20-s + 1.75·23-s + 0.782·25-s + 0.355·26-s − 0.249·28-s − 0.611·29-s + 1.34·31-s + 1.02·32-s − 0.622·34-s − 0.504·35-s − 1.44·37-s − 0.230·38-s + 1.29·40-s − 0.842·41-s + 1.44·43-s + 1.01·46-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 \( 1 - 0.823T + 2T^{2} \)
5 \( 1 + 2.98T + 5T^{2} \)
13 \( 1 - 2.20T + 13T^{2} \)
17 \( 1 + 4.40T + 17T^{2} \)
19 \( 1 + 1.72T + 19T^{2} \)
23 \( 1 - 8.39T + 23T^{2} \)
29 \( 1 + 3.29T + 29T^{2} \)
31 \( 1 - 7.47T + 31T^{2} \)
37 \( 1 + 8.78T + 37T^{2} \)
41 \( 1 + 5.39T + 41T^{2} \)
43 \( 1 - 9.44T + 43T^{2} \)
47 \( 1 - 5.39T + 47T^{2} \)
53 \( 1 + 9.39T + 53T^{2} \)
59 \( 1 + 3.47T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 - 4.32T + 67T^{2} \)
71 \( 1 + 4.40T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 7.18T + 79T^{2} \)
83 \( 1 + 7.63T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 2.85T + 97T^{2} \)
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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

−7.50636476970555516208949140077, −6.85001121977700711012980445196, −6.09596813458985085858751713694, −5.10804824512119760648076539889, −4.66862225508484290447135747638, −3.95277471070987153244283923722, −3.45211543402027363857502811474, −2.51598232586754833141601771693, −1.06249706614282958018771635714, 0, 1.06249706614282958018771635714, 2.51598232586754833141601771693, 3.45211543402027363857502811474, 3.95277471070987153244283923722, 4.66862225508484290447135747638, 5.10804824512119760648076539889, 6.09596813458985085858751713694, 6.85001121977700711012980445196, 7.50636476970555516208949140077

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