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 − 2.75·5-s − 6-s + 0.824·7-s + 8-s + 9-s − 2.75·10-s − 4.01·11-s − 12-s − 13-s + 0.824·14-s + 2.75·15-s + 16-s − 4.88·17-s + 18-s + 5.85·19-s − 2.75·20-s − 0.824·21-s − 4.01·22-s + 6.04·23-s − 24-s + 2.58·25-s − 26-s − 27-s + 0.824·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.23·5-s − 0.408·6-s + 0.311·7-s + 0.353·8-s + 0.333·9-s − 0.870·10-s − 1.21·11-s − 0.288·12-s − 0.277·13-s + 0.220·14-s + 0.710·15-s + 0.250·16-s − 1.18·17-s + 0.235·18-s + 1.34·19-s − 0.615·20-s − 0.179·21-s − 0.856·22-s + 1.26·23-s − 0.204·24-s + 0.516·25-s − 0.196·26-s − 0.192·27-s + 0.155·28-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

\(L(1)\) \(\approx\) \(1.390253903\)
\(L(\frac12)\) \(\approx\) \(1.390253903\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.75T + 5T^{2} \)
7 \( 1 - 0.824T + 7T^{2} \)
11 \( 1 + 4.01T + 11T^{2} \)
17 \( 1 + 4.88T + 17T^{2} \)
19 \( 1 - 5.85T + 19T^{2} \)
23 \( 1 - 6.04T + 23T^{2} \)
29 \( 1 + 6.86T + 29T^{2} \)
31 \( 1 - 0.898T + 31T^{2} \)
37 \( 1 - 4.98T + 37T^{2} \)
41 \( 1 + 8.56T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 + 6.44T + 53T^{2} \)
59 \( 1 + 2.07T + 59T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 - 9.95T + 67T^{2} \)
71 \( 1 - 5.23T + 71T^{2} \)
73 \( 1 + 2.85T + 73T^{2} \)
79 \( 1 - 8.39T + 79T^{2} \)
83 \( 1 + 5.21T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 - 6.12T + 97T^{2} \)
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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

−7.66152611011937149310538547208, −7.21432928470119423694460591045, −6.43724386188676673530515487708, −5.54772606538799829252018979309, −4.86281593553949588603314595157, −4.56976077466199443128732606432, −3.52915408325409430580725654319, −2.94315418481120199911955793279, −1.85305271079765532035561169618, −0.52380225088939975512816934183, 0.52380225088939975512816934183, 1.85305271079765532035561169618, 2.94315418481120199911955793279, 3.52915408325409430580725654319, 4.56976077466199443128732606432, 4.86281593553949588603314595157, 5.54772606538799829252018979309, 6.43724386188676673530515487708, 7.21432928470119423694460591045, 7.66152611011937149310538547208

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