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 + 2.36·5-s − 6-s − 4.26·7-s − 8-s + 9-s − 2.36·10-s − 1.66·11-s + 12-s − 13-s + 4.26·14-s + 2.36·15-s + 16-s + 1.26·17-s − 18-s − 4.50·19-s + 2.36·20-s − 4.26·21-s + 1.66·22-s + 8.71·23-s − 24-s + 0.580·25-s + 26-s + 27-s − 4.26·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.05·5-s − 0.408·6-s − 1.61·7-s − 0.353·8-s + 0.333·9-s − 0.746·10-s − 0.502·11-s + 0.288·12-s − 0.277·13-s + 1.13·14-s + 0.609·15-s + 0.250·16-s + 0.307·17-s − 0.235·18-s − 1.03·19-s + 0.528·20-s − 0.929·21-s + 0.355·22-s + 1.81·23-s − 0.204·24-s + 0.116·25-s + 0.196·26-s + 0.192·27-s − 0.805·28-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 \( 1 - 2.36T + 5T^{2} \)
7 \( 1 + 4.26T + 7T^{2} \)
11 \( 1 + 1.66T + 11T^{2} \)
17 \( 1 - 1.26T + 17T^{2} \)
19 \( 1 + 4.50T + 19T^{2} \)
23 \( 1 - 8.71T + 23T^{2} \)
29 \( 1 - 7.70T + 29T^{2} \)
31 \( 1 + 7.02T + 31T^{2} \)
37 \( 1 - 1.42T + 37T^{2} \)
41 \( 1 + 2.88T + 41T^{2} \)
43 \( 1 + 2.87T + 43T^{2} \)
47 \( 1 - 2.32T + 47T^{2} \)
53 \( 1 + 3.14T + 53T^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 + 4.27T + 61T^{2} \)
67 \( 1 + 9.62T + 67T^{2} \)
71 \( 1 + 7.37T + 71T^{2} \)
73 \( 1 + 5.92T + 73T^{2} \)
79 \( 1 - 5.42T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 - 8.92T + 97T^{2} \)
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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

−7.35061125294277680483254012075, −6.91345284210374282668694304399, −6.24979056765980520282641875256, −5.64297144781018032057047606365, −4.71907560699107651661473100262, −3.55647449882894083703299596162, −2.85598512775288158305526742246, −2.35586675860125948721941227045, −1.27542452568797168325844931754, 0, 1.27542452568797168325844931754, 2.35586675860125948721941227045, 2.85598512775288158305526742246, 3.55647449882894083703299596162, 4.71907560699107651661473100262, 5.64297144781018032057047606365, 6.24979056765980520282641875256, 6.91345284210374282668694304399, 7.35061125294277680483254012075

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