Properties

Label 2-8036-1.1-c1-0-126
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  + 3.15·3-s − 2.22·5-s + 6.96·9-s − 4.12·11-s − 2.30·13-s − 7.00·15-s + 2.24·17-s − 1.92·19-s − 2.59·23-s − 0.0707·25-s + 12.5·27-s + 4.54·29-s − 0.881·31-s − 13.0·33-s + 7.08·37-s − 7.29·39-s + 41-s − 12.1·43-s − 15.4·45-s + 5.83·47-s + 7.09·51-s − 8.01·53-s + 9.15·55-s − 6.07·57-s − 11.7·59-s − 7.46·61-s + 5.12·65-s + ⋯
L(s)  = 1  + 1.82·3-s − 0.992·5-s + 2.32·9-s − 1.24·11-s − 0.640·13-s − 1.80·15-s + 0.545·17-s − 0.441·19-s − 0.541·23-s − 0.0141·25-s + 2.41·27-s + 0.844·29-s − 0.158·31-s − 2.26·33-s + 1.16·37-s − 1.16·39-s + 0.156·41-s − 1.84·43-s − 2.30·45-s + 0.850·47-s + 0.993·51-s − 1.10·53-s + 1.23·55-s − 0.804·57-s − 1.52·59-s − 0.955·61-s + 0.636·65-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 \( 1 - 3.15T + 3T^{2} \)
5 \( 1 + 2.22T + 5T^{2} \)
11 \( 1 + 4.12T + 11T^{2} \)
13 \( 1 + 2.30T + 13T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + 1.92T + 19T^{2} \)
23 \( 1 + 2.59T + 23T^{2} \)
29 \( 1 - 4.54T + 29T^{2} \)
31 \( 1 + 0.881T + 31T^{2} \)
37 \( 1 - 7.08T + 37T^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 - 5.83T + 47T^{2} \)
53 \( 1 + 8.01T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 7.46T + 61T^{2} \)
67 \( 1 - 1.58T + 67T^{2} \)
71 \( 1 - 8.75T + 71T^{2} \)
73 \( 1 + 2.18T + 73T^{2} \)
79 \( 1 + 0.403T + 79T^{2} \)
83 \( 1 + 5.26T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 3.49T + 97T^{2} \)
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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

−7.74249791056340733028566227320, −7.22799976971891831326170270787, −6.29240827397697626227122944724, −5.11230053462910329579716435618, −4.42024638772576192191918774891, −3.78809686779369081071192804054, −2.96531497656159093282948440609, −2.55732862846182159948281807894, −1.53269711813484536901314386714, 0, 1.53269711813484536901314386714, 2.55732862846182159948281807894, 2.96531497656159093282948440609, 3.78809686779369081071192804054, 4.42024638772576192191918774891, 5.11230053462910329579716435618, 6.29240827397697626227122944724, 7.22799976971891831326170270787, 7.74249791056340733028566227320

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