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  − 1.36·2-s − 0.141·4-s − 3.14·5-s + 7-s + 2.91·8-s + 4.28·10-s + 4.77·13-s − 1.36·14-s − 3.69·16-s − 4.77·17-s + 7.00·19-s + 0.443·20-s − 5.14·23-s + 4.86·25-s − 6.51·26-s − 0.141·28-s + 7.00·29-s − 3.63·31-s − 0.797·32-s + 6.51·34-s − 3.14·35-s − 9.86·37-s − 9.55·38-s − 9.17·40-s − 3.22·41-s − 4.28·43-s + 7.00·46-s + ⋯
L(s)  = 1  − 0.964·2-s − 0.0706·4-s − 1.40·5-s + 0.377·7-s + 1.03·8-s + 1.35·10-s + 1.32·13-s − 0.364·14-s − 0.924·16-s − 1.15·17-s + 1.60·19-s + 0.0992·20-s − 1.07·23-s + 0.973·25-s − 1.27·26-s − 0.0267·28-s + 1.30·29-s − 0.653·31-s − 0.141·32-s + 1.11·34-s − 0.530·35-s − 1.62·37-s − 1.55·38-s − 1.45·40-s − 0.503·41-s − 0.653·43-s + 1.03·46-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 \( 1 + 1.36T + 2T^{2} \)
5 \( 1 + 3.14T + 5T^{2} \)
13 \( 1 - 4.77T + 13T^{2} \)
17 \( 1 + 4.77T + 17T^{2} \)
19 \( 1 - 7.00T + 19T^{2} \)
23 \( 1 + 5.14T + 23T^{2} \)
29 \( 1 - 7.00T + 29T^{2} \)
31 \( 1 + 3.63T + 31T^{2} \)
37 \( 1 + 9.86T + 37T^{2} \)
41 \( 1 + 3.22T + 41T^{2} \)
43 \( 1 + 4.28T + 43T^{2} \)
47 \( 1 - 0.778T + 47T^{2} \)
53 \( 1 + 2.28T + 53T^{2} \)
59 \( 1 - 0.363T + 59T^{2} \)
61 \( 1 + 3.22T + 61T^{2} \)
67 \( 1 + 6.59T + 67T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 - 3.22T + 73T^{2} \)
79 \( 1 - 3.71T + 79T^{2} \)
83 \( 1 - 1.55T + 83T^{2} \)
89 \( 1 - 5.58T + 89T^{2} \)
97 \( 1 - 6.15T + 97T^{2} \)
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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

−7.74964893516527937149628308276, −7.12058810388824924021724994207, −6.41687336283557190778323553272, −5.27710038355053337136107573841, −4.61652567720975334695143655297, −3.86055606185165122179937970553, −3.33183148283014022041928849014, −1.90172089076921706656830620604, −0.991645112262633184237523442496, 0, 0.991645112262633184237523442496, 1.90172089076921706656830620604, 3.33183148283014022041928849014, 3.86055606185165122179937970553, 4.61652567720975334695143655297, 5.27710038355053337136107573841, 6.41687336283557190778323553272, 7.12058810388824924021724994207, 7.74964893516527937149628308276

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