Properties

Label 2-6160-1.1-c1-0-2
Degree $2$
Conductor $6160$
Sign $1$
Analytic cond. $49.1878$
Root an. cond. $7.01340$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.73·3-s − 5-s − 7-s + 4.46·9-s − 11-s − 1.46·13-s + 2.73·15-s − 3.46·17-s − 6.73·19-s + 2.73·21-s + 8.19·23-s + 25-s − 3.99·27-s − 4.73·29-s − 2·31-s + 2.73·33-s + 35-s + 0.732·37-s + 4·39-s − 2.19·41-s − 2·43-s − 4.46·45-s − 6.92·47-s + 49-s + 9.46·51-s − 7.26·53-s + 55-s + ⋯
L(s)  = 1  − 1.57·3-s − 0.447·5-s − 0.377·7-s + 1.48·9-s − 0.301·11-s − 0.406·13-s + 0.705·15-s − 0.840·17-s − 1.54·19-s + 0.596·21-s + 1.70·23-s + 0.200·25-s − 0.769·27-s − 0.878·29-s − 0.359·31-s + 0.475·33-s + 0.169·35-s + 0.120·37-s + 0.640·39-s − 0.342·41-s − 0.304·43-s − 0.665·45-s − 1.01·47-s + 0.142·49-s + 1.32·51-s − 0.998·53-s + 0.134·55-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: no
Arithmetic: yes
Character: $\chi_{6160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2664885750\)
\(L(\frac12)\) \(\approx\) \(0.2664885750\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73T + 3T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 6.73T + 19T^{2} \)
23 \( 1 - 8.19T + 23T^{2} \)
29 \( 1 + 4.73T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 0.732T + 37T^{2} \)
41 \( 1 + 2.19T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 7.26T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 + 4.92T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 9.46T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 - 3.46T + 89T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−7.88564588145823208291388660786, −7.11166102552308387219708627411, −6.54884408557104318721660470282, −6.05801450004200455731709333561, −5.03997177873655003403914594033, −4.74744442373279420810105582594, −3.83952629954115571878934888528, −2.78787195514632618009735907954, −1.61313355181238393416060098770, −0.29080531880755536608360242344, 0.29080531880755536608360242344, 1.61313355181238393416060098770, 2.78787195514632618009735907954, 3.83952629954115571878934888528, 4.74744442373279420810105582594, 5.03997177873655003403914594033, 6.05801450004200455731709333561, 6.54884408557104318721660470282, 7.11166102552308387219708627411, 7.88564588145823208291388660786

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