Properties

Label 2-3380-1.1-c1-0-48
Degree $2$
Conductor $3380$
Sign $1$
Analytic cond. $26.9894$
Root an. cond. $5.19513$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5-s − 3·7-s + 6·9-s − 3·11-s + 3·15-s − 7·17-s − 19-s + 9·21-s − 7·23-s + 25-s − 9·27-s − 5·29-s + 4·31-s + 9·33-s + 3·35-s + 3·37-s − 7·41-s − 9·43-s − 6·45-s − 8·47-s + 2·49-s + 21·51-s − 6·53-s + 3·55-s + 3·57-s − 5·59-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s − 1.13·7-s + 2·9-s − 0.904·11-s + 0.774·15-s − 1.69·17-s − 0.229·19-s + 1.96·21-s − 1.45·23-s + 1/5·25-s − 1.73·27-s − 0.928·29-s + 0.718·31-s + 1.56·33-s + 0.507·35-s + 0.493·37-s − 1.09·41-s − 1.37·43-s − 0.894·45-s − 1.16·47-s + 2/7·49-s + 2.94·51-s − 0.824·53-s + 0.404·55-s + 0.397·57-s − 0.650·59-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + p T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−7.71092894854614279464215606279, −6.75269754169353239414511173489, −6.42316901832940664209631111827, −5.71150251702973222489203372128, −4.81463940533850542819919266630, −4.22903446387228037310782146310, −3.14137628761080127354664282251, −1.83951790940148279304958234655, 0, 0, 1.83951790940148279304958234655, 3.14137628761080127354664282251, 4.22903446387228037310782146310, 4.81463940533850542819919266630, 5.71150251702973222489203372128, 6.42316901832940664209631111827, 6.75269754169353239414511173489, 7.71092894854614279464215606279

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