Properties

Degree $2$
Conductor $8036$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.05·3-s − 1.02·5-s − 1.89·9-s − 1.28·11-s − 2.29·13-s + 1.08·15-s + 4.44·17-s − 2.95·19-s + 4.81·23-s − 3.94·25-s + 5.14·27-s + 0.100·29-s + 0.925·31-s + 1.35·33-s − 7.07·37-s + 2.41·39-s + 41-s − 5.47·43-s + 1.95·45-s − 2.99·47-s − 4.67·51-s − 1.46·53-s + 1.32·55-s + 3.10·57-s + 4.37·59-s − 5.39·61-s + 2.36·65-s + ⋯
L(s)  = 1  − 0.606·3-s − 0.460·5-s − 0.631·9-s − 0.387·11-s − 0.637·13-s + 0.279·15-s + 1.07·17-s − 0.676·19-s + 1.00·23-s − 0.788·25-s + 0.990·27-s + 0.0187·29-s + 0.166·31-s + 0.235·33-s − 1.16·37-s + 0.386·39-s + 0.156·41-s − 0.834·43-s + 0.290·45-s − 0.437·47-s − 0.654·51-s − 0.200·53-s + 0.178·55-s + 0.410·57-s + 0.569·59-s − 0.690·61-s + 0.293·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8036\)    =    \(2^{2} \cdot 7^{2} \cdot 41\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8036} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8036,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7215862536\)
\(L(\frac12)\) \(\approx\) \(0.7215862536\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 - T \)
good3 \( 1 + 1.05T + 3T^{2} \)
5 \( 1 + 1.02T + 5T^{2} \)
11 \( 1 + 1.28T + 11T^{2} \)
13 \( 1 + 2.29T + 13T^{2} \)
17 \( 1 - 4.44T + 17T^{2} \)
19 \( 1 + 2.95T + 19T^{2} \)
23 \( 1 - 4.81T + 23T^{2} \)
29 \( 1 - 0.100T + 29T^{2} \)
31 \( 1 - 0.925T + 31T^{2} \)
37 \( 1 + 7.07T + 37T^{2} \)
43 \( 1 + 5.47T + 43T^{2} \)
47 \( 1 + 2.99T + 47T^{2} \)
53 \( 1 + 1.46T + 53T^{2} \)
59 \( 1 - 4.37T + 59T^{2} \)
61 \( 1 + 5.39T + 61T^{2} \)
67 \( 1 + 5.18T + 67T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 + 9.89T + 73T^{2} \)
79 \( 1 + 7.39T + 79T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 - 3.20T + 97T^{2} \)
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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

−7.72444242401781594088660886642, −7.21989528099057869034138085868, −6.37511799108568699422127776154, −5.70288620582030524216699761618, −5.08989879092242047781680507372, −4.45750705080062749943902546268, −3.40421919666076116487251496734, −2.82723245365249182891365418333, −1.68725511538766507601537673522, −0.42589246309817497616644209910, 0.42589246309817497616644209910, 1.68725511538766507601537673522, 2.82723245365249182891365418333, 3.40421919666076116487251496734, 4.45750705080062749943902546268, 5.08989879092242047781680507372, 5.70288620582030524216699761618, 6.37511799108568699422127776154, 7.21989528099057869034138085868, 7.72444242401781594088660886642

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