Properties

Label 2-8036-1.1-c1-0-129
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  + 2.28·3-s + 0.350·5-s + 2.23·9-s + 3.11·11-s − 3.97·13-s + 0.801·15-s − 2.70·17-s − 0.682·19-s − 4.11·23-s − 4.87·25-s − 1.74·27-s − 6.74·29-s + 0.465·31-s + 7.11·33-s − 4.33·37-s − 9.09·39-s + 41-s − 2.58·43-s + 0.783·45-s + 5.96·47-s − 6.18·51-s + 2.04·53-s + 1.08·55-s − 1.56·57-s − 10.3·59-s + 2.87·61-s − 1.39·65-s + ⋯
L(s)  = 1  + 1.32·3-s + 0.156·5-s + 0.746·9-s + 0.937·11-s − 1.10·13-s + 0.206·15-s − 0.655·17-s − 0.156·19-s − 0.858·23-s − 0.975·25-s − 0.335·27-s − 1.25·29-s + 0.0835·31-s + 1.23·33-s − 0.713·37-s − 1.45·39-s + 0.156·41-s − 0.394·43-s + 0.116·45-s + 0.870·47-s − 0.865·51-s + 0.280·53-s + 0.146·55-s − 0.206·57-s − 1.35·59-s + 0.367·61-s − 0.172·65-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 \( 1 - 2.28T + 3T^{2} \)
5 \( 1 - 0.350T + 5T^{2} \)
11 \( 1 - 3.11T + 11T^{2} \)
13 \( 1 + 3.97T + 13T^{2} \)
17 \( 1 + 2.70T + 17T^{2} \)
19 \( 1 + 0.682T + 19T^{2} \)
23 \( 1 + 4.11T + 23T^{2} \)
29 \( 1 + 6.74T + 29T^{2} \)
31 \( 1 - 0.465T + 31T^{2} \)
37 \( 1 + 4.33T + 37T^{2} \)
43 \( 1 + 2.58T + 43T^{2} \)
47 \( 1 - 5.96T + 47T^{2} \)
53 \( 1 - 2.04T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 2.87T + 61T^{2} \)
67 \( 1 + 3.15T + 67T^{2} \)
71 \( 1 + 3.23T + 71T^{2} \)
73 \( 1 + 0.599T + 73T^{2} \)
79 \( 1 + 0.856T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 + 3.40T + 97T^{2} \)
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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

−7.55542516834965651869144694309, −7.02236961513498994477782064917, −6.15590167335115749170414728471, −5.42048758459603692699263487718, −4.34597489349817341221998411757, −3.89631248080611500142325620608, −3.04901519186649027345077020051, −2.19686919485782070783503028282, −1.69361825329777957033844833901, 0, 1.69361825329777957033844833901, 2.19686919485782070783503028282, 3.04901519186649027345077020051, 3.89631248080611500142325620608, 4.34597489349817341221998411757, 5.42048758459603692699263487718, 6.15590167335115749170414728471, 7.02236961513498994477782064917, 7.55542516834965651869144694309

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