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 + 1.34·5-s − 6-s + 2.71·7-s − 8-s + 9-s − 1.34·10-s − 2.39·11-s + 12-s − 13-s − 2.71·14-s + 1.34·15-s + 16-s − 6.50·17-s − 18-s − 0.431·19-s + 1.34·20-s + 2.71·21-s + 2.39·22-s − 4.50·23-s − 24-s − 3.18·25-s + 26-s + 27-s + 2.71·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.602·5-s − 0.408·6-s + 1.02·7-s − 0.353·8-s + 0.333·9-s − 0.426·10-s − 0.721·11-s + 0.288·12-s − 0.277·13-s − 0.725·14-s + 0.348·15-s + 0.250·16-s − 1.57·17-s − 0.235·18-s − 0.0990·19-s + 0.301·20-s + 0.592·21-s + 0.509·22-s − 0.939·23-s − 0.204·24-s − 0.636·25-s + 0.196·26-s + 0.192·27-s + 0.512·28-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 \( 1 - 1.34T + 5T^{2} \)
7 \( 1 - 2.71T + 7T^{2} \)
11 \( 1 + 2.39T + 11T^{2} \)
17 \( 1 + 6.50T + 17T^{2} \)
19 \( 1 + 0.431T + 19T^{2} \)
23 \( 1 + 4.50T + 23T^{2} \)
29 \( 1 + 1.56T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 6.36T + 37T^{2} \)
41 \( 1 + 5.06T + 41T^{2} \)
43 \( 1 - 5.51T + 43T^{2} \)
47 \( 1 + 1.41T + 47T^{2} \)
53 \( 1 + 8.06T + 53T^{2} \)
59 \( 1 - 6.88T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 - 8.38T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + 1.57T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 5.72T + 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + 6.79T + 97T^{2} \)
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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

−7.86377321071031222103250330722, −6.84810283109483160971303225849, −6.33003354679389427284527547091, −5.36104127499960236267610495333, −4.70830536038775164450940973265, −3.90153578692991043937904788377, −2.67780877270026819205030489962, −2.16524147815551817599955623821, −1.47944960258153677638518245169, 0, 1.47944960258153677638518245169, 2.16524147815551817599955623821, 2.67780877270026819205030489962, 3.90153578692991043937904788377, 4.70830536038775164450940973265, 5.36104127499960236267610495333, 6.33003354679389427284527547091, 6.84810283109483160971303225849, 7.86377321071031222103250330722

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