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 − 3.75·5-s − 6-s + 1.91·7-s − 8-s + 9-s + 3.75·10-s − 4.14·11-s + 12-s − 13-s − 1.91·14-s − 3.75·15-s + 16-s − 4.83·17-s − 18-s + 2.13·19-s − 3.75·20-s + 1.91·21-s + 4.14·22-s + 2.08·23-s − 24-s + 9.07·25-s + 26-s + 27-s + 1.91·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.67·5-s − 0.408·6-s + 0.723·7-s − 0.353·8-s + 0.333·9-s + 1.18·10-s − 1.25·11-s + 0.288·12-s − 0.277·13-s − 0.511·14-s − 0.968·15-s + 0.250·16-s − 1.17·17-s − 0.235·18-s + 0.489·19-s − 0.839·20-s + 0.417·21-s + 0.884·22-s + 0.434·23-s − 0.204·24-s + 1.81·25-s + 0.196·26-s + 0.192·27-s + 0.361·28-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 \( 1 + 3.75T + 5T^{2} \)
7 \( 1 - 1.91T + 7T^{2} \)
11 \( 1 + 4.14T + 11T^{2} \)
17 \( 1 + 4.83T + 17T^{2} \)
19 \( 1 - 2.13T + 19T^{2} \)
23 \( 1 - 2.08T + 23T^{2} \)
29 \( 1 + 2.19T + 29T^{2} \)
31 \( 1 - 5.15T + 31T^{2} \)
37 \( 1 - 9.63T + 37T^{2} \)
41 \( 1 - 7.88T + 41T^{2} \)
43 \( 1 - 1.60T + 43T^{2} \)
47 \( 1 + 9.35T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 + 7.21T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + 8.07T + 67T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 - 4.11T + 73T^{2} \)
79 \( 1 + 8.08T + 79T^{2} \)
83 \( 1 - 0.397T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−7.58435117305629765212728042026, −7.29876670592252445234126574073, −6.37545026967720291694374991337, −5.19973095624475414292854070032, −4.55222128622469853462770188019, −3.90901128024172871775655960518, −2.88089691741642071754396156239, −2.38658854951526100383703985050, −1.04879240734906847598177560747, 0, 1.04879240734906847598177560747, 2.38658854951526100383703985050, 2.88089691741642071754396156239, 3.90901128024172871775655960518, 4.55222128622469853462770188019, 5.19973095624475414292854070032, 6.37545026967720291694374991337, 7.29876670592252445234126574073, 7.58435117305629765212728042026

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