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  + 0.870·2-s − 1.24·4-s − 0.709·5-s − 7-s − 2.82·8-s − 0.617·10-s + 3.47·13-s − 0.870·14-s + 0.0306·16-s + 1.34·17-s − 0.503·19-s + 0.882·20-s + 3.69·23-s − 4.49·25-s + 3.02·26-s + 1.24·28-s − 9.21·29-s + 1.39·31-s + 5.67·32-s + 1.17·34-s + 0.709·35-s + 2.24·37-s − 0.437·38-s + 2.00·40-s + 12.4·41-s − 5.60·43-s + 3.21·46-s + ⋯
L(s)  = 1  + 0.615·2-s − 0.621·4-s − 0.317·5-s − 0.377·7-s − 0.997·8-s − 0.195·10-s + 0.962·13-s − 0.232·14-s + 0.00767·16-s + 0.326·17-s − 0.115·19-s + 0.197·20-s + 0.769·23-s − 0.899·25-s + 0.592·26-s + 0.234·28-s − 1.71·29-s + 0.249·31-s + 1.00·32-s + 0.200·34-s + 0.119·35-s + 0.369·37-s − 0.0710·38-s + 0.316·40-s + 1.94·41-s − 0.854·43-s + 0.473·46-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 \( 1 - 0.870T + 2T^{2} \)
5 \( 1 + 0.709T + 5T^{2} \)
13 \( 1 - 3.47T + 13T^{2} \)
17 \( 1 - 1.34T + 17T^{2} \)
19 \( 1 + 0.503T + 19T^{2} \)
23 \( 1 - 3.69T + 23T^{2} \)
29 \( 1 + 9.21T + 29T^{2} \)
31 \( 1 - 1.39T + 31T^{2} \)
37 \( 1 - 2.24T + 37T^{2} \)
41 \( 1 - 12.4T + 41T^{2} \)
43 \( 1 + 5.60T + 43T^{2} \)
47 \( 1 + 5.79T + 47T^{2} \)
53 \( 1 - 3.60T + 53T^{2} \)
59 \( 1 + 7.64T + 59T^{2} \)
61 \( 1 + 5.52T + 61T^{2} \)
67 \( 1 - 3.25T + 67T^{2} \)
71 \( 1 - 8.53T + 71T^{2} \)
73 \( 1 - 2.35T + 73T^{2} \)
79 \( 1 - 4.00T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + 1.42T + 89T^{2} \)
97 \( 1 + 8.56T + 97T^{2} \)
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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

−7.67483589002209469921766410266, −6.61937147411750694740315678131, −6.02064101280000223637035591449, −5.42000262102512973854631484078, −4.64289282538004663967666274160, −3.77994618523085351439280076600, −3.51339801658782036124493640440, −2.47326748524991650489704820275, −1.17124046046932608003922132603, 0, 1.17124046046932608003922132603, 2.47326748524991650489704820275, 3.51339801658782036124493640440, 3.77994618523085351439280076600, 4.64289282538004663967666274160, 5.42000262102512973854631484078, 6.02064101280000223637035591449, 6.61937147411750694740315678131, 7.67483589002209469921766410266

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