Properties

Label 2-8036-1.1-c1-0-84
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  − 2.71·3-s + 0.521·5-s + 4.38·9-s − 0.0523·11-s + 5.95·13-s − 1.41·15-s − 8.06·17-s − 4.41·19-s + 5.28·23-s − 4.72·25-s − 3.76·27-s + 4.73·29-s + 8.03·31-s + 0.142·33-s − 5.28·37-s − 16.1·39-s − 41-s + 3.65·43-s + 2.28·45-s − 4.87·47-s + 21.9·51-s − 1.13·53-s − 0.0273·55-s + 11.9·57-s − 5.21·59-s − 10.2·61-s + 3.10·65-s + ⋯
L(s)  = 1  − 1.56·3-s + 0.233·5-s + 1.46·9-s − 0.0157·11-s + 1.65·13-s − 0.365·15-s − 1.95·17-s − 1.01·19-s + 1.10·23-s − 0.945·25-s − 0.725·27-s + 0.879·29-s + 1.44·31-s + 0.0247·33-s − 0.868·37-s − 2.59·39-s − 0.156·41-s + 0.557·43-s + 0.340·45-s − 0.710·47-s + 3.06·51-s − 0.155·53-s − 0.00368·55-s + 1.58·57-s − 0.679·59-s − 1.31·61-s + 0.385·65-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 \( 1 + 2.71T + 3T^{2} \)
5 \( 1 - 0.521T + 5T^{2} \)
11 \( 1 + 0.0523T + 11T^{2} \)
13 \( 1 - 5.95T + 13T^{2} \)
17 \( 1 + 8.06T + 17T^{2} \)
19 \( 1 + 4.41T + 19T^{2} \)
23 \( 1 - 5.28T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 - 8.03T + 31T^{2} \)
37 \( 1 + 5.28T + 37T^{2} \)
43 \( 1 - 3.65T + 43T^{2} \)
47 \( 1 + 4.87T + 47T^{2} \)
53 \( 1 + 1.13T + 53T^{2} \)
59 \( 1 + 5.21T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 - 9.37T + 67T^{2} \)
71 \( 1 - 9.25T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + 7.46T + 79T^{2} \)
83 \( 1 + 6.66T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 8.41T + 97T^{2} \)
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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

−7.13158452980921428855694536734, −6.47543800516061978475692381657, −6.25768069870924709350867258598, −5.55128263040304285177057178556, −4.53985431263904882641377155097, −4.37212266278320910293990026851, −3.14985900308355988221027396721, −1.98609619005362648591970505689, −1.06856883179536101224375088155, 0, 1.06856883179536101224375088155, 1.98609619005362648591970505689, 3.14985900308355988221027396721, 4.37212266278320910293990026851, 4.53985431263904882641377155097, 5.55128263040304285177057178556, 6.25768069870924709350867258598, 6.47543800516061978475692381657, 7.13158452980921428855694536734

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