Properties

Label 2-7623-1.1-c1-0-271
Degree $2$
Conductor $7623$
Sign $-1$
Analytic cond. $60.8699$
Root an. cond. $7.80192$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.79·2-s + 5.80·4-s − 1.62·5-s − 7-s + 10.6·8-s − 4.54·10-s − 6.23·13-s − 2.79·14-s + 18.1·16-s − 6.00·17-s + 0.305·19-s − 9.44·20-s − 7.84·23-s − 2.35·25-s − 17.4·26-s − 5.80·28-s − 2.94·29-s − 5.31·31-s + 29.3·32-s − 16.7·34-s + 1.62·35-s − 6.41·37-s + 0.854·38-s − 17.2·40-s − 4.04·41-s − 0.640·43-s − 21.9·46-s + ⋯
L(s)  = 1  + 1.97·2-s + 2.90·4-s − 0.727·5-s − 0.377·7-s + 3.76·8-s − 1.43·10-s − 1.72·13-s − 0.746·14-s + 4.52·16-s − 1.45·17-s + 0.0701·19-s − 2.11·20-s − 1.63·23-s − 0.470·25-s − 3.41·26-s − 1.09·28-s − 0.547·29-s − 0.954·31-s + 5.18·32-s − 2.87·34-s + 0.274·35-s − 1.05·37-s + 0.138·38-s − 2.73·40-s − 0.631·41-s − 0.0977·43-s − 3.23·46-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

Degree: \(2\)
Conductor: \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(60.8699\)
Root analytic conductor: \(7.80192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7623,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - 2.79T + 2T^{2} \)
5 \( 1 + 1.62T + 5T^{2} \)
13 \( 1 + 6.23T + 13T^{2} \)
17 \( 1 + 6.00T + 17T^{2} \)
19 \( 1 - 0.305T + 19T^{2} \)
23 \( 1 + 7.84T + 23T^{2} \)
29 \( 1 + 2.94T + 29T^{2} \)
31 \( 1 + 5.31T + 31T^{2} \)
37 \( 1 + 6.41T + 37T^{2} \)
41 \( 1 + 4.04T + 41T^{2} \)
43 \( 1 + 0.640T + 43T^{2} \)
47 \( 1 - 7.86T + 47T^{2} \)
53 \( 1 - 0.251T + 53T^{2} \)
59 \( 1 - 5.46T + 59T^{2} \)
61 \( 1 - 2.45T + 61T^{2} \)
67 \( 1 - 6.54T + 67T^{2} \)
71 \( 1 - 5.61T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 7.16T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 - 9.59T + 89T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.32644245352957963309808713819, −6.71845332699473818294608089175, −5.97980757459619417863636775936, −5.25679877103102599085704085300, −4.62797883839155849811589193107, −3.95228759106331074724299191979, −3.48412472883050264308038814194, −2.35414879769725317781390371831, −2.03351600018119106076756898013, 0, 2.03351600018119106076756898013, 2.35414879769725317781390371831, 3.48412472883050264308038814194, 3.95228759106331074724299191979, 4.62797883839155849811589193107, 5.25679877103102599085704085300, 5.97980757459619417863636775936, 6.71845332699473818294608089175, 7.32644245352957963309808713819

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