Properties

Label 2-6160-1.1-c1-0-20
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  + 5-s − 7-s − 3·9-s + 11-s − 6·13-s − 2·17-s + 4·19-s + 4·23-s + 25-s + 6·29-s − 35-s − 2·37-s − 6·41-s + 4·43-s − 3·45-s − 4·47-s + 49-s − 2·53-s + 55-s − 12·59-s − 2·61-s + 3·63-s − 6·65-s + 8·67-s + 8·71-s − 10·73-s − 77-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 9-s + 0.301·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.169·35-s − 0.328·37-s − 0.937·41-s + 0.609·43-s − 0.447·45-s − 0.583·47-s + 1/7·49-s − 0.274·53-s + 0.134·55-s − 1.56·59-s − 0.256·61-s + 0.377·63-s − 0.744·65-s + 0.977·67-s + 0.949·71-s − 1.17·73-s − 0.113·77-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

\(L(1)\) \(\approx\) \(1.490734432\)
\(L(\frac12)\) \(\approx\) \(1.490734432\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.052970604254017370775256987181, −7.27745281589874855346112476136, −6.64795802357406715157902635794, −5.96756987071107082939343939234, −5.07201203922267525794933283941, −4.72788220773324717822842528107, −3.37373209559994830423666550303, −2.82309501425928002445696560190, −1.98787737800922866892355603523, −0.61392994186852466021014198919, 0.61392994186852466021014198919, 1.98787737800922866892355603523, 2.82309501425928002445696560190, 3.37373209559994830423666550303, 4.72788220773324717822842528107, 5.07201203922267525794933283941, 5.96756987071107082939343939234, 6.64795802357406715157902635794, 7.27745281589874855346112476136, 8.052970604254017370775256987181

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