Properties

Label 2-3380-1.1-c1-0-22
Degree $2$
Conductor $3380$
Sign $-1$
Analytic cond. $26.9894$
Root an. cond. $5.19513$
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  − 2.26·3-s − 5-s − 1.11·7-s + 2.11·9-s − 5.37·11-s + 2.26·15-s + 7.90·19-s + 2.52·21-s + 6.49·23-s + 25-s + 2·27-s + 3.63·29-s − 3.14·31-s + 12.1·33-s + 1.11·35-s − 7.40·37-s − 4.75·41-s + 4.78·43-s − 2.11·45-s + 6.16·47-s − 5.75·49-s + 0.292·53-s + 5.37·55-s − 17.8·57-s + 11.3·59-s − 5.34·61-s − 2.36·63-s + ⋯
L(s)  = 1  − 1.30·3-s − 0.447·5-s − 0.421·7-s + 0.705·9-s − 1.62·11-s + 0.583·15-s + 1.81·19-s + 0.550·21-s + 1.35·23-s + 0.200·25-s + 0.384·27-s + 0.675·29-s − 0.565·31-s + 2.11·33-s + 0.188·35-s − 1.21·37-s − 0.742·41-s + 0.729·43-s − 0.315·45-s + 0.898·47-s − 0.822·49-s + 0.0401·53-s + 0.725·55-s − 2.36·57-s + 1.48·59-s − 0.684·61-s − 0.297·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(26.9894\)
Root analytic conductor: \(5.19513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + 2.26T + 3T^{2} \)
7 \( 1 + 1.11T + 7T^{2} \)
11 \( 1 + 5.37T + 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7.90T + 19T^{2} \)
23 \( 1 - 6.49T + 23T^{2} \)
29 \( 1 - 3.63T + 29T^{2} \)
31 \( 1 + 3.14T + 31T^{2} \)
37 \( 1 + 7.40T + 37T^{2} \)
41 \( 1 + 4.75T + 41T^{2} \)
43 \( 1 - 4.78T + 43T^{2} \)
47 \( 1 - 6.16T + 47T^{2} \)
53 \( 1 - 0.292T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 5.34T + 61T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 - 3.43T + 71T^{2} \)
73 \( 1 + 4.59T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + 7.40T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + 3.76T + 97T^{2} \)
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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

−8.146536950362588763111214233918, −7.23623588741606551383490496820, −6.89604110434204884832650618622, −5.70629146729204416269836977275, −5.31508567230077005618969981159, −4.71545020971716465221253076056, −3.42775434502845963608355215715, −2.70539153527588991816632057526, −1.04606007879802810539920122569, 0, 1.04606007879802810539920122569, 2.70539153527588991816632057526, 3.42775434502845963608355215715, 4.71545020971716465221253076056, 5.31508567230077005618969981159, 5.70629146729204416269836977275, 6.89604110434204884832650618622, 7.23623588741606551383490496820, 8.146536950362588763111214233918

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