Properties

Label 2-6160-1.1-c1-0-15
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  + 0.806·3-s − 5-s − 7-s − 2.35·9-s + 11-s − 1.19·13-s − 0.806·15-s + 1.19·17-s − 4.31·19-s − 0.806·21-s − 4.96·23-s + 25-s − 4.31·27-s − 0.387·29-s + 1.76·31-s + 0.806·33-s + 35-s − 5.35·37-s − 0.962·39-s + 3.76·41-s + 5.73·43-s + 2.35·45-s + 6.73·47-s + 49-s + 0.962·51-s + 4.57·53-s − 55-s + ⋯
L(s)  = 1  + 0.465·3-s − 0.447·5-s − 0.377·7-s − 0.783·9-s + 0.301·11-s − 0.331·13-s − 0.208·15-s + 0.289·17-s − 0.989·19-s − 0.175·21-s − 1.03·23-s + 0.200·25-s − 0.829·27-s − 0.0720·29-s + 0.317·31-s + 0.140·33-s + 0.169·35-s − 0.879·37-s − 0.154·39-s + 0.588·41-s + 0.875·43-s + 0.350·45-s + 0.981·47-s + 0.142·49-s + 0.134·51-s + 0.628·53-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

\(L(1)\) \(\approx\) \(1.420063413\)
\(L(\frac12)\) \(\approx\) \(1.420063413\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.806T + 3T^{2} \)
13 \( 1 + 1.19T + 13T^{2} \)
17 \( 1 - 1.19T + 17T^{2} \)
19 \( 1 + 4.31T + 19T^{2} \)
23 \( 1 + 4.96T + 23T^{2} \)
29 \( 1 + 0.387T + 29T^{2} \)
31 \( 1 - 1.76T + 31T^{2} \)
37 \( 1 + 5.35T + 37T^{2} \)
41 \( 1 - 3.76T + 41T^{2} \)
43 \( 1 - 5.73T + 43T^{2} \)
47 \( 1 - 6.73T + 47T^{2} \)
53 \( 1 - 4.57T + 53T^{2} \)
59 \( 1 - 8.15T + 59T^{2} \)
61 \( 1 - 6.46T + 61T^{2} \)
67 \( 1 - 15.6T + 67T^{2} \)
71 \( 1 + 1.61T + 71T^{2} \)
73 \( 1 + 3.89T + 73T^{2} \)
79 \( 1 - 4.18T + 79T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + 18.6T + 89T^{2} \)
97 \( 1 - 9.53T + 97T^{2} \)
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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

−8.211185316814260132087086114371, −7.41100968859908123918344631336, −6.70299976225798873372309670370, −5.93271543011607764287974294504, −5.27706063284691773290263277335, −4.14901266527034543966773849433, −3.72813402931910702064075661564, −2.73177786358785326512604802623, −2.07929431008458581249781313751, −0.58092057712471202907462826035, 0.58092057712471202907462826035, 2.07929431008458581249781313751, 2.73177786358785326512604802623, 3.72813402931910702064075661564, 4.14901266527034543966773849433, 5.27706063284691773290263277335, 5.93271543011607764287974294504, 6.70299976225798873372309670370, 7.41100968859908123918344631336, 8.211185316814260132087086114371

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