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 − 0.891·5-s + 6-s + 4.07·7-s − 8-s + 9-s + 0.891·10-s − 4.41·11-s − 12-s − 13-s − 4.07·14-s + 0.891·15-s + 16-s − 3.83·17-s − 18-s + 1.31·19-s − 0.891·20-s − 4.07·21-s + 4.41·22-s − 6.87·23-s + 24-s − 4.20·25-s + 26-s − 27-s + 4.07·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.398·5-s + 0.408·6-s + 1.54·7-s − 0.353·8-s + 0.333·9-s + 0.281·10-s − 1.32·11-s − 0.288·12-s − 0.277·13-s − 1.09·14-s + 0.230·15-s + 0.250·16-s − 0.929·17-s − 0.235·18-s + 0.302·19-s − 0.199·20-s − 0.889·21-s + 0.940·22-s − 1.43·23-s + 0.204·24-s − 0.841·25-s + 0.196·26-s − 0.192·27-s + 0.770·28-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

\(L(1)\) \(\approx\) \(0.5958259533\)
\(L(\frac12)\) \(\approx\) \(0.5958259533\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.891T + 5T^{2} \)
7 \( 1 - 4.07T + 7T^{2} \)
11 \( 1 + 4.41T + 11T^{2} \)
17 \( 1 + 3.83T + 17T^{2} \)
19 \( 1 - 1.31T + 19T^{2} \)
23 \( 1 + 6.87T + 23T^{2} \)
29 \( 1 + 7.06T + 29T^{2} \)
31 \( 1 + 4.56T + 31T^{2} \)
37 \( 1 + 8.36T + 37T^{2} \)
41 \( 1 + 0.436T + 41T^{2} \)
43 \( 1 + 3.87T + 43T^{2} \)
47 \( 1 - 1.61T + 47T^{2} \)
53 \( 1 - 4.54T + 53T^{2} \)
59 \( 1 - 3.89T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 3.87T + 67T^{2} \)
71 \( 1 + 8.92T + 71T^{2} \)
73 \( 1 - 9.37T + 73T^{2} \)
79 \( 1 - 8.17T + 79T^{2} \)
83 \( 1 - 4.84T + 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + 1.63T + 97T^{2} \)
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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

−7.76769491240316815559902844081, −7.46124154811833741490504172525, −6.61075226331877944532566758493, −5.48269564212382258440864408427, −5.31941498242415667996724696644, −4.36446188301668761661289231554, −3.59660132185162473601653427421, −2.12861719972307270700483442979, −1.90569829809568410654147462011, −0.41991649614098024565453622333, 0.41991649614098024565453622333, 1.90569829809568410654147462011, 2.12861719972307270700483442979, 3.59660132185162473601653427421, 4.36446188301668761661289231554, 5.31941498242415667996724696644, 5.48269564212382258440864408427, 6.61075226331877944532566758493, 7.46124154811833741490504172525, 7.76769491240316815559902844081

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