Properties

Label 2-8036-1.1-c1-0-104
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  + 1.71·3-s − 2.15·5-s − 0.0659·9-s − 3.02·11-s − 0.848·13-s − 3.68·15-s + 4.76·17-s + 3.90·19-s + 8.25·23-s − 0.372·25-s − 5.25·27-s + 0.0576·29-s − 4.30·31-s − 5.18·33-s + 0.475·37-s − 1.45·39-s − 41-s + 2.82·43-s + 0.141·45-s − 6.71·47-s + 8.15·51-s − 3.42·53-s + 6.51·55-s + 6.68·57-s − 5.72·59-s − 5.96·61-s + 1.82·65-s + ⋯
L(s)  = 1  + 0.988·3-s − 0.962·5-s − 0.0219·9-s − 0.912·11-s − 0.235·13-s − 0.951·15-s + 1.15·17-s + 0.894·19-s + 1.72·23-s − 0.0744·25-s − 1.01·27-s + 0.0107·29-s − 0.772·31-s − 0.902·33-s + 0.0781·37-s − 0.232·39-s − 0.156·41-s + 0.431·43-s + 0.0211·45-s − 0.980·47-s + 1.14·51-s − 0.471·53-s + 0.878·55-s + 0.885·57-s − 0.745·59-s − 0.764·61-s + 0.226·65-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 \( 1 - 1.71T + 3T^{2} \)
5 \( 1 + 2.15T + 5T^{2} \)
11 \( 1 + 3.02T + 11T^{2} \)
13 \( 1 + 0.848T + 13T^{2} \)
17 \( 1 - 4.76T + 17T^{2} \)
19 \( 1 - 3.90T + 19T^{2} \)
23 \( 1 - 8.25T + 23T^{2} \)
29 \( 1 - 0.0576T + 29T^{2} \)
31 \( 1 + 4.30T + 31T^{2} \)
37 \( 1 - 0.475T + 37T^{2} \)
43 \( 1 - 2.82T + 43T^{2} \)
47 \( 1 + 6.71T + 47T^{2} \)
53 \( 1 + 3.42T + 53T^{2} \)
59 \( 1 + 5.72T + 59T^{2} \)
61 \( 1 + 5.96T + 61T^{2} \)
67 \( 1 + 9.53T + 67T^{2} \)
71 \( 1 + 1.84T + 71T^{2} \)
73 \( 1 + 3.89T + 73T^{2} \)
79 \( 1 - 5.11T + 79T^{2} \)
83 \( 1 - 7.41T + 83T^{2} \)
89 \( 1 - 0.305T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−7.58763626506309092459767897481, −7.25671361779976541339371387409, −6.05640789003655280298359225429, −5.25721687900625079451535282025, −4.67452805743869483873551752359, −3.51156791957912783191584415111, −3.25439597505372735466218580004, −2.49965970971102682397102561749, −1.29350661697533612052774485329, 0, 1.29350661697533612052774485329, 2.49965970971102682397102561749, 3.25439597505372735466218580004, 3.51156791957912783191584415111, 4.67452805743869483873551752359, 5.25721687900625079451535282025, 6.05640789003655280298359225429, 7.25671361779976541339371387409, 7.58763626506309092459767897481

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