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 − 1.17·5-s − 6-s − 4.40·7-s + 8-s + 9-s − 1.17·10-s + 0.445·11-s − 12-s − 13-s − 4.40·14-s + 1.17·15-s + 16-s + 6.53·17-s + 18-s − 3.79·19-s − 1.17·20-s + 4.40·21-s + 0.445·22-s − 1.35·23-s − 24-s − 3.62·25-s − 26-s − 27-s − 4.40·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.524·5-s − 0.408·6-s − 1.66·7-s + 0.353·8-s + 0.333·9-s − 0.370·10-s + 0.134·11-s − 0.288·12-s − 0.277·13-s − 1.17·14-s + 0.302·15-s + 0.250·16-s + 1.58·17-s + 0.235·18-s − 0.869·19-s − 0.262·20-s + 0.962·21-s + 0.0950·22-s − 0.282·23-s − 0.204·24-s − 0.725·25-s − 0.196·26-s − 0.192·27-s − 0.833·28-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

\(L(1)\) \(\approx\) \(1.217446101\)
\(L(\frac12)\) \(\approx\) \(1.217446101\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.17T + 5T^{2} \)
7 \( 1 + 4.40T + 7T^{2} \)
11 \( 1 - 0.445T + 11T^{2} \)
17 \( 1 - 6.53T + 17T^{2} \)
19 \( 1 + 3.79T + 19T^{2} \)
23 \( 1 + 1.35T + 23T^{2} \)
29 \( 1 + 5.72T + 29T^{2} \)
31 \( 1 + 7.97T + 31T^{2} \)
37 \( 1 - 6.80T + 37T^{2} \)
41 \( 1 - 0.483T + 41T^{2} \)
43 \( 1 + 7.08T + 43T^{2} \)
47 \( 1 - 3.11T + 47T^{2} \)
53 \( 1 + 2.35T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + 2.55T + 61T^{2} \)
67 \( 1 + 2.47T + 67T^{2} \)
71 \( 1 - 2.87T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + 0.188T + 83T^{2} \)
89 \( 1 - 2.37T + 89T^{2} \)
97 \( 1 + 5.14T + 97T^{2} \)
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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

−7.50505582769120209624823695548, −7.08951243233563486352023861832, −6.24658177206838580591521204792, −5.82710310377976897433480154535, −5.17364524933344910487819529738, −4.03806803018076385918737056082, −3.70772104318278362753843268441, −2.95470196310840070889677268116, −1.87864150229337299857118987498, −0.48365923986594595781424380315, 0.48365923986594595781424380315, 1.87864150229337299857118987498, 2.95470196310840070889677268116, 3.70772104318278362753843268441, 4.03806803018076385918737056082, 5.17364524933344910487819529738, 5.82710310377976897433480154535, 6.24658177206838580591521204792, 7.08951243233563486352023861832, 7.50505582769120209624823695548

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