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  + 1.70·2-s + 0.914·4-s − 4.06·5-s + 7-s − 1.85·8-s − 6.94·10-s + 3.27·13-s + 1.70·14-s − 4.99·16-s + 1.32·17-s + 2.16·19-s − 3.71·20-s + 1.86·23-s + 11.5·25-s + 5.59·26-s + 0.914·28-s + 0.244·29-s − 6.86·31-s − 4.81·32-s + 2.25·34-s − 4.06·35-s + 0.255·37-s + 3.69·38-s + 7.53·40-s + 5.73·41-s − 8.01·43-s + 3.18·46-s + ⋯
L(s)  = 1  + 1.20·2-s + 0.457·4-s − 1.81·5-s + 0.377·7-s − 0.655·8-s − 2.19·10-s + 0.909·13-s + 0.456·14-s − 1.24·16-s + 0.320·17-s + 0.495·19-s − 0.831·20-s + 0.389·23-s + 2.30·25-s + 1.09·26-s + 0.172·28-s + 0.0453·29-s − 1.23·31-s − 0.851·32-s + 0.386·34-s − 0.687·35-s + 0.0420·37-s + 0.598·38-s + 1.19·40-s + 0.895·41-s − 1.22·43-s + 0.469·46-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 \( 1 - 1.70T + 2T^{2} \)
5 \( 1 + 4.06T + 5T^{2} \)
13 \( 1 - 3.27T + 13T^{2} \)
17 \( 1 - 1.32T + 17T^{2} \)
19 \( 1 - 2.16T + 19T^{2} \)
23 \( 1 - 1.86T + 23T^{2} \)
29 \( 1 - 0.244T + 29T^{2} \)
31 \( 1 + 6.86T + 31T^{2} \)
37 \( 1 - 0.255T + 37T^{2} \)
41 \( 1 - 5.73T + 41T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 - 4.06T + 47T^{2} \)
53 \( 1 - 5.00T + 53T^{2} \)
59 \( 1 - 0.983T + 59T^{2} \)
61 \( 1 - 1.84T + 61T^{2} \)
67 \( 1 + 3.00T + 67T^{2} \)
71 \( 1 + 6.47T + 71T^{2} \)
73 \( 1 + 9.65T + 73T^{2} \)
79 \( 1 + 5.53T + 79T^{2} \)
83 \( 1 + 2.07T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + 2.58T + 97T^{2} \)
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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

−7.35493703301229195562697276276, −6.92297985711581151653303988768, −5.89143201619683002111819465753, −5.29052607400497642887977679031, −4.49874734737930414728342420715, −3.97050976050245732993780834551, −3.44612167428702970535010807098, −2.75170159553824904383223892508, −1.23326762595996216353734153920, 0, 1.23326762595996216353734153920, 2.75170159553824904383223892508, 3.44612167428702970535010807098, 3.97050976050245732993780834551, 4.49874734737930414728342420715, 5.29052607400497642887977679031, 5.89143201619683002111819465753, 6.92297985711581151653303988768, 7.35493703301229195562697276276

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