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 − 1.71·5-s − 6-s + 2.07·7-s − 8-s + 9-s + 1.71·10-s − 4.17·11-s + 12-s + 13-s − 2.07·14-s − 1.71·15-s + 16-s − 3.34·17-s − 18-s + 3.21·19-s − 1.71·20-s + 2.07·21-s + 4.17·22-s − 3.16·23-s − 24-s − 2.04·25-s − 26-s + 27-s + 2.07·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.768·5-s − 0.408·6-s + 0.785·7-s − 0.353·8-s + 0.333·9-s + 0.543·10-s − 1.25·11-s + 0.288·12-s + 0.277·13-s − 0.555·14-s − 0.443·15-s + 0.250·16-s − 0.810·17-s − 0.235·18-s + 0.737·19-s − 0.384·20-s + 0.453·21-s + 0.889·22-s − 0.660·23-s − 0.204·24-s − 0.408·25-s − 0.196·26-s + 0.192·27-s + 0.392·28-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 \( 1 + 1.71T + 5T^{2} \)
7 \( 1 - 2.07T + 7T^{2} \)
11 \( 1 + 4.17T + 11T^{2} \)
17 \( 1 + 3.34T + 17T^{2} \)
19 \( 1 - 3.21T + 19T^{2} \)
23 \( 1 + 3.16T + 23T^{2} \)
29 \( 1 - 7.68T + 29T^{2} \)
31 \( 1 + 4.89T + 31T^{2} \)
37 \( 1 - 6.91T + 37T^{2} \)
41 \( 1 - 2.12T + 41T^{2} \)
43 \( 1 - 0.0584T + 43T^{2} \)
47 \( 1 - 8.24T + 47T^{2} \)
53 \( 1 + 4.72T + 53T^{2} \)
59 \( 1 - 3.20T + 59T^{2} \)
61 \( 1 - 1.94T + 61T^{2} \)
67 \( 1 + 8.47T + 67T^{2} \)
71 \( 1 + 1.76T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 - 5.26T + 79T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 + 7.65T + 89T^{2} \)
97 \( 1 + 2.20T + 97T^{2} \)
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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

−7.65514937114524505374456919927, −7.20901104508815471246074341099, −6.22510316843021052935987242939, −5.38102014594434828944470263808, −4.55873279252688034222651260471, −3.87890700626457575997959763888, −2.87195178164202925128416762193, −2.26367317398871337325454746350, −1.21463325897511213215816626106, 0, 1.21463325897511213215816626106, 2.26367317398871337325454746350, 2.87195178164202925128416762193, 3.87890700626457575997959763888, 4.55873279252688034222651260471, 5.38102014594434828944470263808, 6.22510316843021052935987242939, 7.20901104508815471246074341099, 7.65514937114524505374456919927

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