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  + 1.60·2-s + 0.568·4-s + 1.76·5-s − 7-s − 2.29·8-s + 2.82·10-s + 0.237·13-s − 1.60·14-s − 4.81·16-s − 6.14·17-s + 1.53·19-s + 1.00·20-s + 3.89·23-s − 1.88·25-s + 0.381·26-s − 0.568·28-s + 4.45·29-s + 8.52·31-s − 3.12·32-s − 9.84·34-s − 1.76·35-s − 3.64·37-s + 2.46·38-s − 4.04·40-s + 1.90·41-s − 11.7·43-s + 6.24·46-s + ⋯
L(s)  = 1  + 1.13·2-s + 0.284·4-s + 0.789·5-s − 0.377·7-s − 0.811·8-s + 0.894·10-s + 0.0659·13-s − 0.428·14-s − 1.20·16-s − 1.48·17-s + 0.352·19-s + 0.224·20-s + 0.812·23-s − 0.377·25-s + 0.0747·26-s − 0.107·28-s + 0.827·29-s + 1.53·31-s − 0.552·32-s − 1.68·34-s − 0.298·35-s − 0.599·37-s + 0.399·38-s − 0.640·40-s + 0.297·41-s − 1.78·43-s + 0.920·46-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 \( 1 - 1.60T + 2T^{2} \)
5 \( 1 - 1.76T + 5T^{2} \)
13 \( 1 - 0.237T + 13T^{2} \)
17 \( 1 + 6.14T + 17T^{2} \)
19 \( 1 - 1.53T + 19T^{2} \)
23 \( 1 - 3.89T + 23T^{2} \)
29 \( 1 - 4.45T + 29T^{2} \)
31 \( 1 - 8.52T + 31T^{2} \)
37 \( 1 + 3.64T + 37T^{2} \)
41 \( 1 - 1.90T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + 8.55T + 47T^{2} \)
53 \( 1 - 3.14T + 53T^{2} \)
59 \( 1 + 2.56T + 59T^{2} \)
61 \( 1 + 6.78T + 61T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 + 7.37T + 71T^{2} \)
73 \( 1 - 4.60T + 73T^{2} \)
79 \( 1 + 17.2T + 79T^{2} \)
83 \( 1 + 9.16T + 83T^{2} \)
89 \( 1 + 1.14T + 89T^{2} \)
97 \( 1 + 4.68T + 97T^{2} \)
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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

−7.18890934170274945630251420336, −6.47451704031338573334438516558, −6.19380045917005253213414094754, −5.27482809633201625562793920831, −4.74895546887225850765117528286, −4.08309449925607705479628995574, −3.08905671092872305745445390301, −2.60944891874790410328853462814, −1.51285815626070692238415219090, 0, 1.51285815626070692238415219090, 2.60944891874790410328853462814, 3.08905671092872305745445390301, 4.08309449925607705479628995574, 4.74895546887225850765117528286, 5.27482809633201625562793920831, 6.19380045917005253213414094754, 6.47451704031338573334438516558, 7.18890934170274945630251420336

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