Properties

Label 2-8024-1.1-c1-0-10
Degree $2$
Conductor $8024$
Sign $1$
Analytic cond. $64.0719$
Root an. cond. $8.00449$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.45·3-s − 4.36·5-s − 1.77·7-s + 3.03·9-s + 2.04·11-s + 2.44·13-s + 10.7·15-s − 17-s − 3.14·19-s + 4.35·21-s − 5.70·23-s + 14.0·25-s − 0.0785·27-s − 4.70·29-s + 6.94·31-s − 5.01·33-s + 7.73·35-s + 2.42·37-s − 6.00·39-s − 11.8·41-s + 3.44·43-s − 13.2·45-s + 0.480·47-s − 3.84·49-s + 2.45·51-s − 9.20·53-s − 8.89·55-s + ⋯
L(s)  = 1  − 1.41·3-s − 1.94·5-s − 0.670·7-s + 1.01·9-s + 0.615·11-s + 0.678·13-s + 2.76·15-s − 0.242·17-s − 0.722·19-s + 0.951·21-s − 1.18·23-s + 2.80·25-s − 0.0151·27-s − 0.874·29-s + 1.24·31-s − 0.872·33-s + 1.30·35-s + 0.398·37-s − 0.961·39-s − 1.85·41-s + 0.524·43-s − 1.97·45-s + 0.0700·47-s − 0.549·49-s + 0.343·51-s − 1.26·53-s − 1.19·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8024\)    =    \(2^{3} \cdot 17 \cdot 59\)
Sign: $1$
Analytic conductor: \(64.0719\)
Root analytic conductor: \(8.00449\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1849345520\)
\(L(\frac12)\) \(\approx\) \(0.1849345520\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 - T \)
good3 \( 1 + 2.45T + 3T^{2} \)
5 \( 1 + 4.36T + 5T^{2} \)
7 \( 1 + 1.77T + 7T^{2} \)
11 \( 1 - 2.04T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
19 \( 1 + 3.14T + 19T^{2} \)
23 \( 1 + 5.70T + 23T^{2} \)
29 \( 1 + 4.70T + 29T^{2} \)
31 \( 1 - 6.94T + 31T^{2} \)
37 \( 1 - 2.42T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 3.44T + 43T^{2} \)
47 \( 1 - 0.480T + 47T^{2} \)
53 \( 1 + 9.20T + 53T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 + 0.214T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 2.21T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 - 6.52T + 83T^{2} \)
89 \( 1 + 8.56T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−7.85057391686069541226384114485, −6.79934033159172947007111811245, −6.63174226247547911047143552355, −5.87097227918267493330697795275, −4.95862859498622210191717587912, −4.17395607250672558001560133757, −3.85779360205716781592904107376, −2.92466670045996266395200181577, −1.34147051477794643110211268691, −0.24885053351367011125992644327, 0.24885053351367011125992644327, 1.34147051477794643110211268691, 2.92466670045996266395200181577, 3.85779360205716781592904107376, 4.17395607250672558001560133757, 4.95862859498622210191717587912, 5.87097227918267493330697795275, 6.63174226247547911047143552355, 6.79934033159172947007111811245, 7.85057391686069541226384114485

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