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  − 2.39·2-s + 3.71·4-s + 2.58·5-s + 7-s − 4.09·8-s − 6.19·10-s − 6.78·13-s − 2.39·14-s + 2.36·16-s − 3.14·17-s + 1.04·19-s + 9.61·20-s + 6.52·23-s + 1.70·25-s + 16.2·26-s + 3.71·28-s − 0.607·29-s − 8.83·31-s + 2.54·32-s + 7.51·34-s + 2.58·35-s + 8.94·37-s − 2.49·38-s − 10.6·40-s + 8.69·41-s − 4.48·43-s − 15.6·46-s + ⋯
L(s)  = 1  − 1.69·2-s + 1.85·4-s + 1.15·5-s + 0.377·7-s − 1.44·8-s − 1.95·10-s − 1.88·13-s − 0.638·14-s + 0.590·16-s − 0.762·17-s + 0.239·19-s + 2.15·20-s + 1.36·23-s + 0.341·25-s + 3.17·26-s + 0.701·28-s − 0.112·29-s − 1.58·31-s + 0.449·32-s + 1.28·34-s + 0.437·35-s + 1.47·37-s − 0.405·38-s − 1.67·40-s + 1.35·41-s − 0.683·43-s − 2.30·46-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  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9046309990$
$L(\frac12)$  $\approx$  $0.9046309990$
$L(\frac{3}{2})$   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 \( 1 + 2.39T + 2T^{2} \)
5 \( 1 - 2.58T + 5T^{2} \)
13 \( 1 + 6.78T + 13T^{2} \)
17 \( 1 + 3.14T + 17T^{2} \)
19 \( 1 - 1.04T + 19T^{2} \)
23 \( 1 - 6.52T + 23T^{2} \)
29 \( 1 + 0.607T + 29T^{2} \)
31 \( 1 + 8.83T + 31T^{2} \)
37 \( 1 - 8.94T + 37T^{2} \)
41 \( 1 - 8.69T + 41T^{2} \)
43 \( 1 + 4.48T + 43T^{2} \)
47 \( 1 + 3.26T + 47T^{2} \)
53 \( 1 + 1.89T + 53T^{2} \)
59 \( 1 - 0.174T + 59T^{2} \)
61 \( 1 + 8.13T + 61T^{2} \)
67 \( 1 + 7.33T + 67T^{2} \)
71 \( 1 + 2.70T + 71T^{2} \)
73 \( 1 + 0.589T + 73T^{2} \)
79 \( 1 - 9.80T + 79T^{2} \)
83 \( 1 - 8.74T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−7.70263598469090574967527248718, −7.52400403578341343171079036210, −6.73541442759570634283259797407, −6.04744811413946502377735134264, −5.15627327738166369603904010060, −4.54496004859240786454342065767, −2.99214437919731132254633833063, −2.24605934580626206109942644491, −1.74898812712152215982925946153, −0.60466661847827539411606027653, 0.60466661847827539411606027653, 1.74898812712152215982925946153, 2.24605934580626206109942644491, 2.99214437919731132254633833063, 4.54496004859240786454342065767, 5.15627327738166369603904010060, 6.04744811413946502377735134264, 6.73541442759570634283259797407, 7.52400403578341343171079036210, 7.70263598469090574967527248718

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