Properties

Label 2-8037-1.1-c1-0-113
Degree $2$
Conductor $8037$
Sign $-1$
Analytic cond. $64.1757$
Root an. cond. $8.01097$
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  − 0.880·2-s − 1.22·4-s − 0.815·5-s − 4.34·7-s + 2.83·8-s + 0.718·10-s − 4.85·11-s − 3.63·13-s + 3.82·14-s − 0.0480·16-s + 1.61·17-s − 19-s + 0.999·20-s + 4.27·22-s + 4.93·23-s − 4.33·25-s + 3.19·26-s + 5.32·28-s + 3.39·29-s + 10.2·31-s − 5.63·32-s − 1.42·34-s + 3.54·35-s − 8.34·37-s + 0.880·38-s − 2.31·40-s + 10.1·41-s + ⋯
L(s)  = 1  − 0.622·2-s − 0.612·4-s − 0.364·5-s − 1.64·7-s + 1.00·8-s + 0.227·10-s − 1.46·11-s − 1.00·13-s + 1.02·14-s − 0.0120·16-s + 0.391·17-s − 0.229·19-s + 0.223·20-s + 0.911·22-s + 1.02·23-s − 0.866·25-s + 0.626·26-s + 1.00·28-s + 0.630·29-s + 1.83·31-s − 0.996·32-s − 0.243·34-s + 0.599·35-s − 1.37·37-s + 0.142·38-s − 0.366·40-s + 1.58·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8037\)    =    \(3^{2} \cdot 19 \cdot 47\)
Sign: $-1$
Analytic conductor: \(64.1757\)
Root analytic conductor: \(8.01097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8037,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 + T \)
47 \( 1 - T \)
good2 \( 1 + 0.880T + 2T^{2} \)
5 \( 1 + 0.815T + 5T^{2} \)
7 \( 1 + 4.34T + 7T^{2} \)
11 \( 1 + 4.85T + 11T^{2} \)
13 \( 1 + 3.63T + 13T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
23 \( 1 - 4.93T + 23T^{2} \)
29 \( 1 - 3.39T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 8.34T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 4.44T + 43T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 1.63T + 59T^{2} \)
61 \( 1 - 7.54T + 61T^{2} \)
67 \( 1 - 0.210T + 67T^{2} \)
71 \( 1 + 0.376T + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 - 5.55T + 83T^{2} \)
89 \( 1 - 5.74T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−7.67072063572384986119886463993, −6.93466597620286385562939962413, −6.23922835184062791482160061333, −5.23674322640000122108556391653, −4.79857156676391687698041708520, −3.80983268563522955808565930855, −3.04531952951719568406932452289, −2.35930138986939226927774570739, −0.76374860477721521650173999348, 0, 0.76374860477721521650173999348, 2.35930138986939226927774570739, 3.04531952951719568406932452289, 3.80983268563522955808565930855, 4.79857156676391687698041708520, 5.23674322640000122108556391653, 6.23922835184062791482160061333, 6.93466597620286385562939962413, 7.67072063572384986119886463993

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