Properties

Degree $2$
Conductor $8034$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 4.10·5-s − 6-s + 1.08·7-s + 8-s + 9-s + 4.10·10-s + 1.94·11-s − 12-s − 13-s + 1.08·14-s − 4.10·15-s + 16-s + 3.29·17-s + 18-s + 5.56·19-s + 4.10·20-s − 1.08·21-s + 1.94·22-s + 4.23·23-s − 24-s + 11.8·25-s − 26-s − 27-s + 1.08·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.83·5-s − 0.408·6-s + 0.408·7-s + 0.353·8-s + 0.333·9-s + 1.29·10-s + 0.585·11-s − 0.288·12-s − 0.277·13-s + 0.289·14-s − 1.06·15-s + 0.250·16-s + 0.797·17-s + 0.235·18-s + 1.27·19-s + 0.918·20-s − 0.235·21-s + 0.414·22-s + 0.883·23-s − 0.204·24-s + 2.37·25-s − 0.196·26-s − 0.192·27-s + 0.204·28-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: \(0\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.964928305\)
\(L(\frac12)\) \(\approx\) \(4.964928305\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.10T + 5T^{2} \)
7 \( 1 - 1.08T + 7T^{2} \)
11 \( 1 - 1.94T + 11T^{2} \)
17 \( 1 - 3.29T + 17T^{2} \)
19 \( 1 - 5.56T + 19T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 - 5.73T + 29T^{2} \)
31 \( 1 + 9.76T + 31T^{2} \)
37 \( 1 + 0.721T + 37T^{2} \)
41 \( 1 - 2.73T + 41T^{2} \)
43 \( 1 + 1.00T + 43T^{2} \)
47 \( 1 + 7.53T + 47T^{2} \)
53 \( 1 - 6.31T + 53T^{2} \)
59 \( 1 - 5.34T + 59T^{2} \)
61 \( 1 + 9.12T + 61T^{2} \)
67 \( 1 + 3.91T + 67T^{2} \)
71 \( 1 + 3.70T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 7.86T + 79T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 - 3.37T + 89T^{2} \)
97 \( 1 + 3.56T + 97T^{2} \)
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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

−7.47826002989464357568888171956, −6.96303236747466135193438419428, −6.22700282477532383865724982109, −5.64460300202300183317933822403, −5.18992001765753434073346431096, −4.62731679538698413502926530159, −3.43387394673974060227776280615, −2.70864619541163769387815079702, −1.65882942640313884379779524639, −1.17733499268567198749081238160, 1.17733499268567198749081238160, 1.65882942640313884379779524639, 2.70864619541163769387815079702, 3.43387394673974060227776280615, 4.62731679538698413502926530159, 5.18992001765753434073346431096, 5.64460300202300183317933822403, 6.22700282477532383865724982109, 6.96303236747466135193438419428, 7.47826002989464357568888171956

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