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  + 2.43·2-s + 3.90·4-s − 1.25·5-s + 7-s + 4.63·8-s − 3.05·10-s − 3.17·13-s + 2.43·14-s + 3.45·16-s − 5.92·17-s + 2.86·19-s − 4.91·20-s − 6.76·23-s − 3.41·25-s − 7.72·26-s + 3.90·28-s − 4.49·29-s + 9.72·31-s − 0.882·32-s − 14.3·34-s − 1.25·35-s − 5.45·37-s + 6.95·38-s − 5.83·40-s + 0.314·41-s − 0.132·43-s − 16.4·46-s + ⋯
L(s)  = 1  + 1.71·2-s + 1.95·4-s − 0.562·5-s + 0.377·7-s + 1.63·8-s − 0.967·10-s − 0.881·13-s + 0.649·14-s + 0.862·16-s − 1.43·17-s + 0.656·19-s − 1.09·20-s − 1.41·23-s − 0.683·25-s − 1.51·26-s + 0.738·28-s − 0.834·29-s + 1.74·31-s − 0.155·32-s − 2.46·34-s − 0.212·35-s − 0.896·37-s + 1.12·38-s − 0.922·40-s + 0.0491·41-s − 0.0202·43-s − 2.42·46-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 \( 1 - 2.43T + 2T^{2} \)
5 \( 1 + 1.25T + 5T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
17 \( 1 + 5.92T + 17T^{2} \)
19 \( 1 - 2.86T + 19T^{2} \)
23 \( 1 + 6.76T + 23T^{2} \)
29 \( 1 + 4.49T + 29T^{2} \)
31 \( 1 - 9.72T + 31T^{2} \)
37 \( 1 + 5.45T + 37T^{2} \)
41 \( 1 - 0.314T + 41T^{2} \)
43 \( 1 + 0.132T + 43T^{2} \)
47 \( 1 + 9.37T + 47T^{2} \)
53 \( 1 + 4.35T + 53T^{2} \)
59 \( 1 - 6.94T + 59T^{2} \)
61 \( 1 + 2.45T + 61T^{2} \)
67 \( 1 + 9.41T + 67T^{2} \)
71 \( 1 - 0.116T + 71T^{2} \)
73 \( 1 - 0.615T + 73T^{2} \)
79 \( 1 - 8.52T + 79T^{2} \)
83 \( 1 + 0.950T + 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−7.38530680892873553101352502482, −6.59040586629317185857666251823, −6.06184097399708454752926507810, −5.17598903275637206669147391581, −4.64860999574286969432459457314, −4.08420796175314911792338334297, −3.36805351232760595997369310585, −2.46454065330327922120755902304, −1.80473489381734329220556980305, 0, 1.80473489381734329220556980305, 2.46454065330327922120755902304, 3.36805351232760595997369310585, 4.08420796175314911792338334297, 4.64860999574286969432459457314, 5.17598903275637206669147391581, 6.06184097399708454752926507810, 6.59040586629317185857666251823, 7.38530680892873553101352502482

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