Properties

Degree $2$
Conductor $8034$
Sign $-1$
Motivic weight $1$
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 + 0.290·5-s − 6-s − 3.15·7-s − 8-s + 9-s − 0.290·10-s + 5.57·11-s + 12-s − 13-s + 3.15·14-s + 0.290·15-s + 16-s − 2.13·17-s − 18-s − 5.05·19-s + 0.290·20-s − 3.15·21-s − 5.57·22-s + 1.96·23-s − 24-s − 4.91·25-s + 26-s + 27-s − 3.15·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.129·5-s − 0.408·6-s − 1.19·7-s − 0.353·8-s + 0.333·9-s − 0.0919·10-s + 1.68·11-s + 0.288·12-s − 0.277·13-s + 0.843·14-s + 0.0750·15-s + 0.250·16-s − 0.518·17-s − 0.235·18-s − 1.15·19-s + 0.0649·20-s − 0.688·21-s − 1.18·22-s + 0.410·23-s − 0.204·24-s − 0.983·25-s + 0.196·26-s + 0.192·27-s − 0.596·28-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: \(1\)
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 \( 1 - 0.290T + 5T^{2} \)
7 \( 1 + 3.15T + 7T^{2} \)
11 \( 1 - 5.57T + 11T^{2} \)
17 \( 1 + 2.13T + 17T^{2} \)
19 \( 1 + 5.05T + 19T^{2} \)
23 \( 1 - 1.96T + 23T^{2} \)
29 \( 1 + 0.444T + 29T^{2} \)
31 \( 1 - 2.60T + 31T^{2} \)
37 \( 1 + 7.55T + 37T^{2} \)
41 \( 1 - 8.43T + 41T^{2} \)
43 \( 1 + 1.60T + 43T^{2} \)
47 \( 1 - 9.45T + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 0.462T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + 8.17T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + 7.96T + 89T^{2} \)
97 \( 1 - 17.9T + 97T^{2} \)
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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

−7.37016470354776054078188769891, −6.90630667279814734011737530929, −6.31826604185087046974613137238, −5.72848577214127904096579297700, −4.30217661990212031993868702600, −3.89245405377752096247136978793, −2.95596773388151833021471189298, −2.19917920367049794069490500641, −1.26544681117673891306055167251, 0, 1.26544681117673891306055167251, 2.19917920367049794069490500641, 2.95596773388151833021471189298, 3.89245405377752096247136978793, 4.30217661990212031993868702600, 5.72848577214127904096579297700, 6.31826604185087046974613137238, 6.90630667279814734011737530929, 7.37016470354776054078188769891

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