Properties

Label 2-8036-1.1-c1-0-106
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.30·3-s + 3.30·5-s + 7.90·9-s + 11-s + 5·13-s + 10.9·15-s − 6.21·17-s − 3.90·19-s − 3.30·23-s + 5.90·25-s + 16.2·27-s + 10.3·29-s − 0.0916·31-s + 3.30·33-s − 0.605·37-s + 16.5·39-s + 41-s − 11.6·43-s + 26.1·45-s + 5.39·47-s − 20.5·51-s − 2.69·53-s + 3.30·55-s − 12.9·57-s − 5.90·59-s + 10.2·61-s + 16.5·65-s + ⋯
L(s)  = 1  + 1.90·3-s + 1.47·5-s + 2.63·9-s + 0.301·11-s + 1.38·13-s + 2.81·15-s − 1.50·17-s − 0.896·19-s − 0.688·23-s + 1.18·25-s + 3.11·27-s + 1.91·29-s − 0.0164·31-s + 0.574·33-s − 0.0995·37-s + 2.64·39-s + 0.156·41-s − 1.76·43-s + 3.89·45-s + 0.786·47-s − 2.87·51-s − 0.370·53-s + 0.445·55-s − 1.70·57-s − 0.769·59-s + 1.30·61-s + 2.04·65-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: \(0\)
Selberg data: \((2,\ 8036,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.466185138\)
\(L(\frac12)\) \(\approx\) \(6.466185138\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.30T + 3T^{2} \)
5 \( 1 - 3.30T + 5T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + 6.21T + 17T^{2} \)
19 \( 1 + 3.90T + 19T^{2} \)
23 \( 1 + 3.30T + 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + 0.0916T + 31T^{2} \)
37 \( 1 + 0.605T + 37T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 - 5.39T + 47T^{2} \)
53 \( 1 + 2.69T + 53T^{2} \)
59 \( 1 + 5.90T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 - 1.90T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 - 8.51T + 89T^{2} \)
97 \( 1 + 9.11T + 97T^{2} \)
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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

−8.188321110831117079718253589436, −7.09821661694762481092699688293, −6.46972429020839511798112126250, −6.06424743798278492719366442306, −4.75066671987062043245986007123, −4.18394956186334251089442632284, −3.34228521515358622384660519204, −2.53635105582756006745090762891, −1.96087733388379321563108560913, −1.31086234224679349344492469723, 1.31086234224679349344492469723, 1.96087733388379321563108560913, 2.53635105582756006745090762891, 3.34228521515358622384660519204, 4.18394956186334251089442632284, 4.75066671987062043245986007123, 6.06424743798278492719366442306, 6.46972429020839511798112126250, 7.09821661694762481092699688293, 8.188321110831117079718253589436

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