Properties

Label 2-8036-1.1-c1-0-128
Degree $2$
Conductor $8036$
Sign $-1$
Analytic cond. $64.1677$
Root an. cond. $8.01047$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.86·3-s − 1.20·5-s + 5.22·9-s − 0.847·11-s + 1.77·13-s − 3.44·15-s − 1.33·17-s − 5.12·19-s − 0.816·23-s − 3.55·25-s + 6.36·27-s − 8.14·29-s − 5.88·31-s − 2.42·33-s + 1.65·37-s + 5.09·39-s − 41-s − 3.82·43-s − 6.27·45-s − 4.67·47-s − 3.81·51-s − 3.64·53-s + 1.01·55-s − 14.6·57-s + 0.851·59-s + 9.13·61-s − 2.13·65-s + ⋯
L(s)  = 1  + 1.65·3-s − 0.537·5-s + 1.74·9-s − 0.255·11-s + 0.492·13-s − 0.889·15-s − 0.322·17-s − 1.17·19-s − 0.170·23-s − 0.711·25-s + 1.22·27-s − 1.51·29-s − 1.05·31-s − 0.422·33-s + 0.272·37-s + 0.815·39-s − 0.156·41-s − 0.583·43-s − 0.935·45-s − 0.681·47-s − 0.534·51-s − 0.501·53-s + 0.137·55-s − 1.94·57-s + 0.110·59-s + 1.16·61-s − 0.264·65-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: $-1$
Analytic conductor: \(64.1677\)
Root analytic conductor: \(8.01047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8036,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 - 2.86T + 3T^{2} \)
5 \( 1 + 1.20T + 5T^{2} \)
11 \( 1 + 0.847T + 11T^{2} \)
13 \( 1 - 1.77T + 13T^{2} \)
17 \( 1 + 1.33T + 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
23 \( 1 + 0.816T + 23T^{2} \)
29 \( 1 + 8.14T + 29T^{2} \)
31 \( 1 + 5.88T + 31T^{2} \)
37 \( 1 - 1.65T + 37T^{2} \)
43 \( 1 + 3.82T + 43T^{2} \)
47 \( 1 + 4.67T + 47T^{2} \)
53 \( 1 + 3.64T + 53T^{2} \)
59 \( 1 - 0.851T + 59T^{2} \)
61 \( 1 - 9.13T + 61T^{2} \)
67 \( 1 + 8.39T + 67T^{2} \)
71 \( 1 - 1.66T + 71T^{2} \)
73 \( 1 + 3.08T + 73T^{2} \)
79 \( 1 - 8.12T + 79T^{2} \)
83 \( 1 + 3.90T + 83T^{2} \)
89 \( 1 - 9.52T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−7.70153682093312907382654791073, −7.06049195277225823620075470955, −6.25163326586163728172120237161, −5.31470840278884621438739172841, −4.25516582417236055124219091736, −3.82717393796372414264892618471, −3.18594246070939521299373051815, −2.22812907272853874843382396641, −1.67409145253713901857800686911, 0, 1.67409145253713901857800686911, 2.22812907272853874843382396641, 3.18594246070939521299373051815, 3.82717393796372414264892618471, 4.25516582417236055124219091736, 5.31470840278884621438739172841, 6.25163326586163728172120237161, 7.06049195277225823620075470955, 7.70153682093312907382654791073

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