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 − 2.57·5-s − 6-s − 4.81·7-s − 8-s + 9-s + 2.57·10-s + 1.68·11-s + 12-s − 13-s + 4.81·14-s − 2.57·15-s + 16-s + 6.18·17-s − 18-s + 0.771·19-s − 2.57·20-s − 4.81·21-s − 1.68·22-s − 2.46·23-s − 24-s + 1.61·25-s + 26-s + 27-s − 4.81·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.15·5-s − 0.408·6-s − 1.81·7-s − 0.353·8-s + 0.333·9-s + 0.813·10-s + 0.506·11-s + 0.288·12-s − 0.277·13-s + 1.28·14-s − 0.664·15-s + 0.250·16-s + 1.49·17-s − 0.235·18-s + 0.176·19-s − 0.575·20-s − 1.05·21-s − 0.358·22-s − 0.513·23-s − 0.204·24-s + 0.323·25-s + 0.196·26-s + 0.192·27-s − 0.909·28-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 \( 1 + 2.57T + 5T^{2} \)
7 \( 1 + 4.81T + 7T^{2} \)
11 \( 1 - 1.68T + 11T^{2} \)
17 \( 1 - 6.18T + 17T^{2} \)
19 \( 1 - 0.771T + 19T^{2} \)
23 \( 1 + 2.46T + 23T^{2} \)
29 \( 1 + 4.09T + 29T^{2} \)
31 \( 1 + 7.36T + 31T^{2} \)
37 \( 1 - 7.10T + 37T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 - 4.87T + 43T^{2} \)
47 \( 1 + 7.45T + 47T^{2} \)
53 \( 1 - 4.54T + 53T^{2} \)
59 \( 1 + 5.91T + 59T^{2} \)
61 \( 1 + 0.0881T + 61T^{2} \)
67 \( 1 + 9.81T + 67T^{2} \)
71 \( 1 + 6.15T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 + 3.92T + 79T^{2} \)
83 \( 1 + 1.22T + 83T^{2} \)
89 \( 1 - 6.45T + 89T^{2} \)
97 \( 1 + 8.09T + 97T^{2} \)
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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

−7.51518700566527926179778611806, −7.17512795077838866761794970973, −6.18790844689412588354932227330, −5.72008324008686738316394211041, −4.32143172631233108310964227148, −3.58553753459048357442804218170, −3.26482777101139502590695638719, −2.32986902366311676861853963242, −0.989210729909969253617787556756, 0, 0.989210729909969253617787556756, 2.32986902366311676861853963242, 3.26482777101139502590695638719, 3.58553753459048357442804218170, 4.32143172631233108310964227148, 5.72008324008686738316394211041, 6.18790844689412588354932227330, 7.17512795077838866761794970973, 7.51518700566527926179778611806

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