Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s + 0.618·4-s + 4.23·5-s − 7-s + 2.23·8-s − 6.85·10-s + 6.23·13-s + 1.61·14-s − 4.85·16-s + 4.47·17-s + 3·19-s + 2.61·20-s − 8.47·23-s + 12.9·25-s − 10.0·26-s − 0.618·28-s + 3·29-s + 3.38·32-s − 7.23·34-s − 4.23·35-s + 3.47·37-s − 4.85·38-s + 9.47·40-s − 1.52·41-s + 10.9·43-s + 13.7·46-s + 3·47-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.309·4-s + 1.89·5-s − 0.377·7-s + 0.790·8-s − 2.16·10-s + 1.72·13-s + 0.432·14-s − 1.21·16-s + 1.08·17-s + 0.688·19-s + 0.585·20-s − 1.76·23-s + 2.58·25-s − 1.97·26-s − 0.116·28-s + 0.557·29-s + 0.597·32-s − 1.24·34-s − 0.716·35-s + 0.570·37-s − 0.787·38-s + 1.49·40-s − 0.238·41-s + 1.66·43-s + 2.02·46-s + 0.437·47-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  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.858428451$
$L(\frac12)$  $\approx$  $1.858428451$
$L(\frac{3}{2})$   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 \( 1 + 1.61T + 2T^{2} \)
5 \( 1 - 4.23T + 5T^{2} \)
13 \( 1 - 6.23T + 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + 8.47T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 3.47T + 37T^{2} \)
41 \( 1 + 1.52T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 8.94T + 53T^{2} \)
59 \( 1 - 1.47T + 59T^{2} \)
61 \( 1 - 3.52T + 61T^{2} \)
67 \( 1 + 8.70T + 67T^{2} \)
71 \( 1 + 1.52T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.054889660924422808795874581962, −7.33614294437518082727250367668, −6.31375161914847456974448985304, −6.01770258814134310729353519639, −5.37132773687808546506323146494, −4.32029451758007965253316838943, −3.34848217830436975198088244269, −2.34574996071983003778520550714, −1.49227785884528830042038683493, −0.921392421691580142330643156260, 0.921392421691580142330643156260, 1.49227785884528830042038683493, 2.34574996071983003778520550714, 3.34848217830436975198088244269, 4.32029451758007965253316838943, 5.37132773687808546506323146494, 6.01770258814134310729353519639, 6.31375161914847456974448985304, 7.33614294437518082727250367668, 8.054889660924422808795874581962

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