Properties

Degree $2$
Conductor $8034$
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-s − 3-s + 4-s − 2.05·5-s − 6-s − 1.63·7-s + 8-s + 9-s − 2.05·10-s − 1.96·11-s − 12-s − 13-s − 1.63·14-s + 2.05·15-s + 16-s − 4.75·17-s + 18-s − 7.72·19-s − 2.05·20-s + 1.63·21-s − 1.96·22-s − 0.952·23-s − 24-s − 0.793·25-s − 26-s − 27-s − 1.63·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.917·5-s − 0.408·6-s − 0.618·7-s + 0.353·8-s + 0.333·9-s − 0.648·10-s − 0.593·11-s − 0.288·12-s − 0.277·13-s − 0.437·14-s + 0.529·15-s + 0.250·16-s − 1.15·17-s + 0.235·18-s − 1.77·19-s − 0.458·20-s + 0.356·21-s − 0.419·22-s − 0.198·23-s − 0.204·24-s − 0.158·25-s − 0.196·26-s − 0.192·27-s − 0.309·28-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: $1$
Motivic weight: \(1\)
Character: $\chi_{8034} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8559124749\)
\(L(\frac12)\) \(\approx\) \(0.8559124749\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.05T + 5T^{2} \)
7 \( 1 + 1.63T + 7T^{2} \)
11 \( 1 + 1.96T + 11T^{2} \)
17 \( 1 + 4.75T + 17T^{2} \)
19 \( 1 + 7.72T + 19T^{2} \)
23 \( 1 + 0.952T + 23T^{2} \)
29 \( 1 + 2.09T + 29T^{2} \)
31 \( 1 - 9.14T + 31T^{2} \)
37 \( 1 - 2.77T + 37T^{2} \)
41 \( 1 + 9.56T + 41T^{2} \)
43 \( 1 - 3.74T + 43T^{2} \)
47 \( 1 - 5.58T + 47T^{2} \)
53 \( 1 + 1.37T + 53T^{2} \)
59 \( 1 + 9.37T + 59T^{2} \)
61 \( 1 - 2.60T + 61T^{2} \)
67 \( 1 - 5.60T + 67T^{2} \)
71 \( 1 + 0.768T + 71T^{2} \)
73 \( 1 - 9.15T + 73T^{2} \)
79 \( 1 - 3.23T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + 6.28T + 89T^{2} \)
97 \( 1 + 7.59T + 97T^{2} \)
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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

−7.83569019908431977886640621746, −6.76653511038633496432868408345, −6.56662906632620935746491302138, −5.78425967230533607396633988761, −4.84252155150469664505253552492, −4.35564771568118407844644011271, −3.73634075430750110605518509575, −2.75807948997682341802546048170, −1.99238242522553977958332705547, −0.39608150294583736725460033027, 0.39608150294583736725460033027, 1.99238242522553977958332705547, 2.75807948997682341802546048170, 3.73634075430750110605518509575, 4.35564771568118407844644011271, 4.84252155150469664505253552492, 5.78425967230533607396633988761, 6.56662906632620935746491302138, 6.76653511038633496432868408345, 7.83569019908431977886640621746

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