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 − 0.503·5-s + 6-s + 2.75·7-s − 8-s + 9-s + 0.503·10-s + 6.03·11-s − 12-s − 13-s − 2.75·14-s + 0.503·15-s + 16-s − 6.29·17-s − 18-s − 6.31·19-s − 0.503·20-s − 2.75·21-s − 6.03·22-s − 7.22·23-s + 24-s − 4.74·25-s + 26-s − 27-s + 2.75·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.225·5-s + 0.408·6-s + 1.04·7-s − 0.353·8-s + 0.333·9-s + 0.159·10-s + 1.81·11-s − 0.288·12-s − 0.277·13-s − 0.737·14-s + 0.130·15-s + 0.250·16-s − 1.52·17-s − 0.235·18-s − 1.44·19-s − 0.112·20-s − 0.602·21-s − 1.28·22-s − 1.50·23-s + 0.204·24-s − 0.949·25-s + 0.196·26-s − 0.192·27-s + 0.521·28-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

\(L(1)\) \(\approx\) \(1.083184612\)
\(L(\frac12)\) \(\approx\) \(1.083184612\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.503T + 5T^{2} \)
7 \( 1 - 2.75T + 7T^{2} \)
11 \( 1 - 6.03T + 11T^{2} \)
17 \( 1 + 6.29T + 17T^{2} \)
19 \( 1 + 6.31T + 19T^{2} \)
23 \( 1 + 7.22T + 23T^{2} \)
29 \( 1 - 9.35T + 29T^{2} \)
31 \( 1 - 2.92T + 31T^{2} \)
37 \( 1 + 2.31T + 37T^{2} \)
41 \( 1 + 3.89T + 41T^{2} \)
43 \( 1 - 4.29T + 43T^{2} \)
47 \( 1 + 6.99T + 47T^{2} \)
53 \( 1 - 3.51T + 53T^{2} \)
59 \( 1 + 9.24T + 59T^{2} \)
61 \( 1 + 0.178T + 61T^{2} \)
67 \( 1 - 9.08T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 4.54T + 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 0.865T + 83T^{2} \)
89 \( 1 - 0.262T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−8.095712576564663723684061107658, −7.04697141809968252732503111253, −6.37235307163411462265412626093, −6.21573958254799578692433332498, −4.84894299533633670776498373784, −4.37641527331804506052429887676, −3.71269688490918243605167461290, −2.17250319558394160532226284575, −1.75934998831297184820209086181, −0.59389553833442049994122446076, 0.59389553833442049994122446076, 1.75934998831297184820209086181, 2.17250319558394160532226284575, 3.71269688490918243605167461290, 4.37641527331804506052429887676, 4.84894299533633670776498373784, 6.21573958254799578692433332498, 6.37235307163411462265412626093, 7.04697141809968252732503111253, 8.095712576564663723684061107658

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