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 + 3.69·5-s − 6-s − 0.801·7-s − 8-s + 9-s − 3.69·10-s − 2.09·11-s + 12-s − 13-s + 0.801·14-s + 3.69·15-s + 16-s − 7.28·17-s − 18-s + 6.41·19-s + 3.69·20-s − 0.801·21-s + 2.09·22-s − 6.18·23-s − 24-s + 8.61·25-s + 26-s + 27-s − 0.801·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.65·5-s − 0.408·6-s − 0.303·7-s − 0.353·8-s + 0.333·9-s − 1.16·10-s − 0.631·11-s + 0.288·12-s − 0.277·13-s + 0.214·14-s + 0.952·15-s + 0.250·16-s − 1.76·17-s − 0.235·18-s + 1.47·19-s + 0.825·20-s − 0.174·21-s + 0.446·22-s − 1.28·23-s − 0.204·24-s + 1.72·25-s + 0.196·26-s + 0.192·27-s − 0.151·28-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 \( 1 - 3.69T + 5T^{2} \)
7 \( 1 + 0.801T + 7T^{2} \)
11 \( 1 + 2.09T + 11T^{2} \)
17 \( 1 + 7.28T + 17T^{2} \)
19 \( 1 - 6.41T + 19T^{2} \)
23 \( 1 + 6.18T + 23T^{2} \)
29 \( 1 + 8.16T + 29T^{2} \)
31 \( 1 + 4.92T + 31T^{2} \)
37 \( 1 - 7.56T + 37T^{2} \)
41 \( 1 + 12.6T + 41T^{2} \)
43 \( 1 + 1.69T + 43T^{2} \)
47 \( 1 + 1.40T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 - 3.80T + 61T^{2} \)
67 \( 1 + 1.82T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + 9.57T + 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 + 9.35T + 83T^{2} \)
89 \( 1 - 3.88T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−7.36829555644634527565325836905, −7.02507448562932173470481123281, −6.07953624937111098644566289827, −5.62456156038229431667106823859, −4.81032084342939105334674192357, −3.68524906723285175434290655783, −2.73205190708534080686985930305, −2.13612695106904058515930715364, −1.54447951269332685682479836098, 0, 1.54447951269332685682479836098, 2.13612695106904058515930715364, 2.73205190708534080686985930305, 3.68524906723285175434290655783, 4.81032084342939105334674192357, 5.62456156038229431667106823859, 6.07953624937111098644566289827, 7.02507448562932173470481123281, 7.36829555644634527565325836905

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