Properties

Label 2-8034-1.1-c1-0-185
Degree $2$
Conductor $8034$
Sign $-1$
Analytic cond. $64.1518$
Root an. cond. $8.00948$
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-s − 3-s + 4-s + 1.44·5-s − 6-s + 0.367·7-s + 8-s + 9-s + 1.44·10-s + 0.547·11-s − 12-s + 13-s + 0.367·14-s − 1.44·15-s + 16-s − 3.55·17-s + 18-s − 4.08·19-s + 1.44·20-s − 0.367·21-s + 0.547·22-s − 6.44·23-s − 24-s − 2.91·25-s + 26-s − 27-s + 0.367·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.645·5-s − 0.408·6-s + 0.138·7-s + 0.353·8-s + 0.333·9-s + 0.456·10-s + 0.165·11-s − 0.288·12-s + 0.277·13-s + 0.0982·14-s − 0.372·15-s + 0.250·16-s − 0.862·17-s + 0.235·18-s − 0.937·19-s + 0.322·20-s − 0.0802·21-s + 0.116·22-s − 1.34·23-s − 0.204·24-s − 0.583·25-s + 0.196·26-s − 0.192·27-s + 0.0694·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
Sign: $-1$
Analytic conductor: \(64.1518\)
Root analytic conductor: \(8.00948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 + T \)
good5 \( 1 - 1.44T + 5T^{2} \)
7 \( 1 - 0.367T + 7T^{2} \)
11 \( 1 - 0.547T + 11T^{2} \)
17 \( 1 + 3.55T + 17T^{2} \)
19 \( 1 + 4.08T + 19T^{2} \)
23 \( 1 + 6.44T + 23T^{2} \)
29 \( 1 - 0.496T + 29T^{2} \)
31 \( 1 - 1.35T + 31T^{2} \)
37 \( 1 + 1.55T + 37T^{2} \)
41 \( 1 + 0.751T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + 7.51T + 47T^{2} \)
53 \( 1 - 1.23T + 53T^{2} \)
59 \( 1 - 3.24T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 7.76T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 - 8.69T + 83T^{2} \)
89 \( 1 + 3.90T + 89T^{2} \)
97 \( 1 - 0.761T + 97T^{2} \)
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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

−7.30730269250499233037839165059, −6.46448330562562726806122354526, −6.06424201476245632081038464571, −5.56405415740169791482018182202, −4.44869670014349503845229272338, −4.29561490028019678161597797105, −3.14778001069062404703597110003, −2.14477704181566443501137935634, −1.54738789717763171414539532389, 0, 1.54738789717763171414539532389, 2.14477704181566443501137935634, 3.14778001069062404703597110003, 4.29561490028019678161597797105, 4.44869670014349503845229272338, 5.56405415740169791482018182202, 6.06424201476245632081038464571, 6.46448330562562726806122354526, 7.30730269250499233037839165059

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