Properties

Label 2-6160-1.1-c1-0-28
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  − 2.77·3-s + 5-s + 7-s + 4.72·9-s + 11-s + 2.77·13-s − 2.77·15-s − 2.02·17-s − 3.55·19-s − 2.77·21-s − 5.72·23-s + 25-s − 4.80·27-s + 8.80·29-s + 7.75·31-s − 2.77·33-s + 35-s + 0.272·37-s − 7.72·39-s − 0.507·41-s − 6.97·43-s + 4.72·45-s + 8.23·47-s + 49-s + 5.62·51-s + 3.72·53-s + 55-s + ⋯
L(s)  = 1  − 1.60·3-s + 0.447·5-s + 0.377·7-s + 1.57·9-s + 0.301·11-s + 0.770·13-s − 0.717·15-s − 0.490·17-s − 0.816·19-s − 0.606·21-s − 1.19·23-s + 0.200·25-s − 0.924·27-s + 1.63·29-s + 1.39·31-s − 0.483·33-s + 0.169·35-s + 0.0447·37-s − 1.23·39-s − 0.0792·41-s − 1.06·43-s + 0.704·45-s + 1.20·47-s + 0.142·49-s + 0.787·51-s + 0.512·53-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.238794502\)
\(L(\frac12)\) \(\approx\) \(1.238794502\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.77T + 3T^{2} \)
13 \( 1 - 2.77T + 13T^{2} \)
17 \( 1 + 2.02T + 17T^{2} \)
19 \( 1 + 3.55T + 19T^{2} \)
23 \( 1 + 5.72T + 23T^{2} \)
29 \( 1 - 8.80T + 29T^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 - 0.272T + 37T^{2} \)
41 \( 1 + 0.507T + 41T^{2} \)
43 \( 1 + 6.97T + 43T^{2} \)
47 \( 1 - 8.23T + 47T^{2} \)
53 \( 1 - 3.72T + 53T^{2} \)
59 \( 1 - 2.19T + 59T^{2} \)
61 \( 1 - 8.06T + 61T^{2} \)
67 \( 1 - 4.48T + 67T^{2} \)
71 \( 1 + 3.45T + 71T^{2} \)
73 \( 1 - 6.02T + 73T^{2} \)
79 \( 1 + 6.14T + 79T^{2} \)
83 \( 1 - 0.802T + 83T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−8.227572195268371136898077985872, −6.92633629735003809742380632809, −6.53924551914888269499762401857, −5.96668842048233595990357197897, −5.33310545117814385948234309820, −4.50452607574538231209102784694, −4.03117880434557497789172245940, −2.61354174796622924308103594880, −1.56595985252228171513110638396, −0.66890659209274585593944103898, 0.66890659209274585593944103898, 1.56595985252228171513110638396, 2.61354174796622924308103594880, 4.03117880434557497789172245940, 4.50452607574538231209102784694, 5.33310545117814385948234309820, 5.96668842048233595990357197897, 6.53924551914888269499762401857, 6.92633629735003809742380632809, 8.227572195268371136898077985872

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