Properties

Label 2-6160-1.1-c1-0-50
Degree $2$
Conductor $6160$
Sign $1$
Analytic cond. $49.1878$
Root an. cond. $7.01340$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s + 7-s + 9-s − 11-s + 2·15-s + 2·21-s + 4·23-s + 25-s − 4·27-s + 2·29-s + 2·31-s − 2·33-s + 35-s − 6·37-s + 8·41-s + 12·43-s + 45-s + 6·47-s + 49-s − 6·53-s − 55-s + 10·59-s − 4·61-s + 63-s + 8·67-s + 8·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.516·15-s + 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.769·27-s + 0.371·29-s + 0.359·31-s − 0.348·33-s + 0.169·35-s − 0.986·37-s + 1.24·41-s + 1.82·43-s + 0.149·45-s + 0.875·47-s + 1/7·49-s − 0.824·53-s − 0.134·55-s + 1.30·59-s − 0.512·61-s + 0.125·63-s + 0.977·67-s + 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6160\)    =    \(2^{4} \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(49.1878\)
Root analytic conductor: \(7.01340\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{6160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.550333811\)
\(L(\frac12)\) \(\approx\) \(3.550333811\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.050211902943267076784687279670, −7.56267747122931163077444789139, −6.78056032363685606232921096912, −5.87997738209408176395348485438, −5.18933435514271702317430389940, −4.33125394928057241468119174674, −3.49279777040919907040866877211, −2.65605437910558752350573792184, −2.12726925429771010968869240672, −0.951657342869011093483118392059, 0.951657342869011093483118392059, 2.12726925429771010968869240672, 2.65605437910558752350573792184, 3.49279777040919907040866877211, 4.33125394928057241468119174674, 5.18933435514271702317430389940, 5.87997738209408176395348485438, 6.78056032363685606232921096912, 7.56267747122931163077444789139, 8.050211902943267076784687279670

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