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 − 2.71·5-s + 6-s − 2.34·7-s − 8-s + 9-s + 2.71·10-s + 0.805·11-s − 12-s − 13-s + 2.34·14-s + 2.71·15-s + 16-s − 7.05·17-s − 18-s + 3.05·19-s − 2.71·20-s + 2.34·21-s − 0.805·22-s − 3.54·23-s + 24-s + 2.36·25-s + 26-s − 27-s − 2.34·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.21·5-s + 0.408·6-s − 0.885·7-s − 0.353·8-s + 0.333·9-s + 0.858·10-s + 0.242·11-s − 0.288·12-s − 0.277·13-s + 0.625·14-s + 0.700·15-s + 0.250·16-s − 1.71·17-s − 0.235·18-s + 0.699·19-s − 0.606·20-s + 0.511·21-s − 0.171·22-s − 0.740·23-s + 0.204·24-s + 0.472·25-s + 0.196·26-s − 0.192·27-s − 0.442·28-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

\(L(1)\) \(\approx\) \(0.03817289577\)
\(L(\frac12)\) \(\approx\) \(0.03817289577\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.71T + 5T^{2} \)
7 \( 1 + 2.34T + 7T^{2} \)
11 \( 1 - 0.805T + 11T^{2} \)
17 \( 1 + 7.05T + 17T^{2} \)
19 \( 1 - 3.05T + 19T^{2} \)
23 \( 1 + 3.54T + 23T^{2} \)
29 \( 1 + 1.98T + 29T^{2} \)
31 \( 1 + 7.06T + 31T^{2} \)
37 \( 1 - 1.93T + 37T^{2} \)
41 \( 1 + 1.08T + 41T^{2} \)
43 \( 1 + 3.41T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 - 6.23T + 53T^{2} \)
59 \( 1 - 1.79T + 59T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 - 4.26T + 67T^{2} \)
71 \( 1 + 0.749T + 71T^{2} \)
73 \( 1 + 4.06T + 73T^{2} \)
79 \( 1 - 4.75T + 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + 7.49T + 89T^{2} \)
97 \( 1 + 9.03T + 97T^{2} \)
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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

−7.85508006125008387349032926978, −7.01116063758136092380181126694, −6.76742229937300588907127651992, −5.93310780162110365023600779128, −5.05923498600809424759956253172, −4.13136918130837017469653634239, −3.60898281707651671114876764607, −2.63523033951027529707240244244, −1.54681861106467407428261788737, −0.10995244218046691468045906696, 0.10995244218046691468045906696, 1.54681861106467407428261788737, 2.63523033951027529707240244244, 3.60898281707651671114876764607, 4.13136918130837017469653634239, 5.05923498600809424759956253172, 5.93310780162110365023600779128, 6.76742229937300588907127651992, 7.01116063758136092380181126694, 7.85508006125008387349032926978

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