Properties

Label 2-8036-1.1-c1-0-28
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  − 2.73·3-s + 0.863·5-s + 4.49·9-s − 5.59·11-s + 1.90·13-s − 2.36·15-s + 4.01·17-s + 1.91·19-s + 9.06·23-s − 4.25·25-s − 4.08·27-s + 5.61·29-s − 9.11·31-s + 15.3·33-s + 1.45·37-s − 5.20·39-s + 41-s + 7.83·43-s + 3.87·45-s − 10.1·47-s − 11.0·51-s − 1.16·53-s − 4.82·55-s − 5.24·57-s − 6.86·59-s − 5.22·61-s + 1.63·65-s + ⋯
L(s)  = 1  − 1.58·3-s + 0.385·5-s + 1.49·9-s − 1.68·11-s + 0.526·13-s − 0.610·15-s + 0.974·17-s + 0.439·19-s + 1.89·23-s − 0.851·25-s − 0.786·27-s + 1.04·29-s − 1.63·31-s + 2.66·33-s + 0.239·37-s − 0.832·39-s + 0.156·41-s + 1.19·43-s + 0.578·45-s − 1.48·47-s − 1.54·51-s − 0.159·53-s − 0.650·55-s − 0.695·57-s − 0.894·59-s − 0.669·61-s + 0.203·65-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

\(L(1)\) \(\approx\) \(0.9974291703\)
\(L(\frac12)\) \(\approx\) \(0.9974291703\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73T + 3T^{2} \)
5 \( 1 - 0.863T + 5T^{2} \)
11 \( 1 + 5.59T + 11T^{2} \)
13 \( 1 - 1.90T + 13T^{2} \)
17 \( 1 - 4.01T + 17T^{2} \)
19 \( 1 - 1.91T + 19T^{2} \)
23 \( 1 - 9.06T + 23T^{2} \)
29 \( 1 - 5.61T + 29T^{2} \)
31 \( 1 + 9.11T + 31T^{2} \)
37 \( 1 - 1.45T + 37T^{2} \)
43 \( 1 - 7.83T + 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + 1.16T + 53T^{2} \)
59 \( 1 + 6.86T + 59T^{2} \)
61 \( 1 + 5.22T + 61T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 - 3.14T + 71T^{2} \)
73 \( 1 - 4.41T + 73T^{2} \)
79 \( 1 + 5.30T + 79T^{2} \)
83 \( 1 + 6.73T + 83T^{2} \)
89 \( 1 - 3.99T + 89T^{2} \)
97 \( 1 + 3.44T + 97T^{2} \)
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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

−7.63672465720574608696840318197, −7.08860553945462235568683536168, −6.24597201746195110885136963940, −5.64258987112025323777263660820, −5.21327759418664937594347451111, −4.70009736235896761019520435861, −3.50806052396108847986661006971, −2.68239795105684109397128014071, −1.47070426323086359813620399343, −0.56809646315956447695941622405, 0.56809646315956447695941622405, 1.47070426323086359813620399343, 2.68239795105684109397128014071, 3.50806052396108847986661006971, 4.70009736235896761019520435861, 5.21327759418664937594347451111, 5.64258987112025323777263660820, 6.24597201746195110885136963940, 7.08860553945462235568683536168, 7.63672465720574608696840318197

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