Properties

Label 2-8036-1.1-c1-0-125
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  + 2.15·3-s + 0.0591·5-s + 1.63·9-s − 5.19·11-s − 1.33·13-s + 0.127·15-s + 5.25·17-s + 6.42·19-s − 5.30·23-s − 4.99·25-s − 2.94·27-s − 2.05·29-s + 4.21·31-s − 11.1·33-s − 6.95·37-s − 2.87·39-s + 41-s − 4.70·43-s + 0.0964·45-s − 4.09·47-s + 11.3·51-s + 2.41·53-s − 0.307·55-s + 13.8·57-s + 4.59·59-s + 4.59·61-s − 0.0790·65-s + ⋯
L(s)  = 1  + 1.24·3-s + 0.0264·5-s + 0.543·9-s − 1.56·11-s − 0.370·13-s + 0.0328·15-s + 1.27·17-s + 1.47·19-s − 1.10·23-s − 0.999·25-s − 0.566·27-s − 0.382·29-s + 0.756·31-s − 1.94·33-s − 1.14·37-s − 0.460·39-s + 0.156·41-s − 0.717·43-s + 0.0143·45-s − 0.597·47-s + 1.58·51-s + 0.331·53-s − 0.0414·55-s + 1.83·57-s + 0.597·59-s + 0.588·61-s − 0.00980·65-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 \( 1 - 2.15T + 3T^{2} \)
5 \( 1 - 0.0591T + 5T^{2} \)
11 \( 1 + 5.19T + 11T^{2} \)
13 \( 1 + 1.33T + 13T^{2} \)
17 \( 1 - 5.25T + 17T^{2} \)
19 \( 1 - 6.42T + 19T^{2} \)
23 \( 1 + 5.30T + 23T^{2} \)
29 \( 1 + 2.05T + 29T^{2} \)
31 \( 1 - 4.21T + 31T^{2} \)
37 \( 1 + 6.95T + 37T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + 4.09T + 47T^{2} \)
53 \( 1 - 2.41T + 53T^{2} \)
59 \( 1 - 4.59T + 59T^{2} \)
61 \( 1 - 4.59T + 61T^{2} \)
67 \( 1 + 9.74T + 67T^{2} \)
71 \( 1 - 2.36T + 71T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
79 \( 1 + 17.6T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + 0.670T + 89T^{2} \)
97 \( 1 - 3.02T + 97T^{2} \)
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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

−7.47594665632227878278745090957, −7.32434799641919510202319805975, −5.84317931975976654799897860631, −5.51307890506739894842907559613, −4.62471575167709369427274169191, −3.58718700405558541670842264594, −3.09305146987166575147714675730, −2.40523097744330703097147545509, −1.52253867369232910440420346194, 0, 1.52253867369232910440420346194, 2.40523097744330703097147545509, 3.09305146987166575147714675730, 3.58718700405558541670842264594, 4.62471575167709369427274169191, 5.51307890506739894842907559613, 5.84317931975976654799897860631, 7.32434799641919510202319805975, 7.47594665632227878278745090957

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