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  − 2.05·2-s + 2.23·4-s + 3.05·5-s + 7-s − 0.492·8-s − 6.29·10-s − 3.32·13-s − 2.05·14-s − 3.46·16-s + 5.37·17-s − 8.29·19-s + 6.84·20-s + 1.74·23-s + 4.35·25-s + 6.84·26-s + 2.23·28-s − 8.16·29-s − 5.44·31-s + 8.11·32-s − 11.0·34-s + 3.05·35-s + 4.53·37-s + 17.0·38-s − 1.50·40-s − 6.84·41-s + 1.84·43-s − 3.58·46-s + ⋯
L(s)  = 1  − 1.45·2-s + 1.11·4-s + 1.36·5-s + 0.377·7-s − 0.174·8-s − 1.99·10-s − 0.922·13-s − 0.550·14-s − 0.866·16-s + 1.30·17-s − 1.90·19-s + 1.53·20-s + 0.363·23-s + 0.871·25-s + 1.34·26-s + 0.423·28-s − 1.51·29-s − 0.978·31-s + 1.43·32-s − 1.89·34-s + 0.517·35-s + 0.745·37-s + 2.77·38-s − 0.238·40-s − 1.06·41-s + 0.282·43-s − 0.528·46-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 \( 1 + 2.05T + 2T^{2} \)
5 \( 1 - 3.05T + 5T^{2} \)
13 \( 1 + 3.32T + 13T^{2} \)
17 \( 1 - 5.37T + 17T^{2} \)
19 \( 1 + 8.29T + 19T^{2} \)
23 \( 1 - 1.74T + 23T^{2} \)
29 \( 1 + 8.16T + 29T^{2} \)
31 \( 1 + 5.44T + 31T^{2} \)
37 \( 1 - 4.53T + 37T^{2} \)
41 \( 1 + 6.84T + 41T^{2} \)
43 \( 1 - 1.84T + 43T^{2} \)
47 \( 1 - 7.28T + 47T^{2} \)
53 \( 1 - 0.985T + 53T^{2} \)
59 \( 1 + 4.52T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 - 0.170T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 + 1.62T + 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 - 3.44T + 97T^{2} \)
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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

−7.50223133787633064314443381007, −7.18417451617224950537883724356, −6.22367844359818865806240770304, −5.61137777783146325121362489394, −4.88245444364321798527231414301, −3.89286044659267053587532369965, −2.54439667864073125802320061953, −2.00955101649199936892504172383, −1.29131866498753778410155147059, 0, 1.29131866498753778410155147059, 2.00955101649199936892504172383, 2.54439667864073125802320061953, 3.89286044659267053587532369965, 4.88245444364321798527231414301, 5.61137777783146325121362489394, 6.22367844359818865806240770304, 7.18417451617224950537883724356, 7.50223133787633064314443381007

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