Properties

Label 2-6040-1.1-c1-0-97
Degree $2$
Conductor $6040$
Sign $1$
Analytic cond. $48.2296$
Root an. cond. $6.94475$
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  + 3.17·3-s + 5-s − 0.749·7-s + 7.09·9-s + 5.92·11-s − 5.85·13-s + 3.17·15-s + 1.49·17-s + 3.53·19-s − 2.38·21-s + 0.831·23-s + 25-s + 13.0·27-s + 5.12·29-s + 8.73·31-s + 18.8·33-s − 0.749·35-s − 6.40·37-s − 18.6·39-s − 5.69·41-s + 6.93·43-s + 7.09·45-s − 2.32·47-s − 6.43·49-s + 4.76·51-s − 9.29·53-s + 5.92·55-s + ⋯
L(s)  = 1  + 1.83·3-s + 0.447·5-s − 0.283·7-s + 2.36·9-s + 1.78·11-s − 1.62·13-s + 0.820·15-s + 0.363·17-s + 0.811·19-s − 0.519·21-s + 0.173·23-s + 0.200·25-s + 2.50·27-s + 0.951·29-s + 1.56·31-s + 3.27·33-s − 0.126·35-s − 1.05·37-s − 2.97·39-s − 0.888·41-s + 1.05·43-s + 1.05·45-s − 0.339·47-s − 0.919·49-s + 0.667·51-s − 1.27·53-s + 0.798·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6040\)    =    \(2^{3} \cdot 5 \cdot 151\)
Sign: $1$
Analytic conductor: \(48.2296\)
Root analytic conductor: \(6.94475\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.982825711\)
\(L(\frac12)\) \(\approx\) \(4.982825711\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
151 \( 1 - T \)
good3 \( 1 - 3.17T + 3T^{2} \)
7 \( 1 + 0.749T + 7T^{2} \)
11 \( 1 - 5.92T + 11T^{2} \)
13 \( 1 + 5.85T + 13T^{2} \)
17 \( 1 - 1.49T + 17T^{2} \)
19 \( 1 - 3.53T + 19T^{2} \)
23 \( 1 - 0.831T + 23T^{2} \)
29 \( 1 - 5.12T + 29T^{2} \)
31 \( 1 - 8.73T + 31T^{2} \)
37 \( 1 + 6.40T + 37T^{2} \)
41 \( 1 + 5.69T + 41T^{2} \)
43 \( 1 - 6.93T + 43T^{2} \)
47 \( 1 + 2.32T + 47T^{2} \)
53 \( 1 + 9.29T + 53T^{2} \)
59 \( 1 - 3.99T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 6.62T + 67T^{2} \)
71 \( 1 - 3.69T + 71T^{2} \)
73 \( 1 - 1.95T + 73T^{2} \)
79 \( 1 - 2.81T + 79T^{2} \)
83 \( 1 + 17.5T + 83T^{2} \)
89 \( 1 + 6.27T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−8.167631029198573615278976332254, −7.40093744349002691279955792953, −6.88536360405237823952775505759, −6.19950398655799805319114481220, −4.87380771512045236080038238066, −4.36490587423552886035306655721, −3.30497511880225008068112298975, −2.95285464620906213175894026413, −1.96699692709095836047221299563, −1.18875532363804678772836889388, 1.18875532363804678772836889388, 1.96699692709095836047221299563, 2.95285464620906213175894026413, 3.30497511880225008068112298975, 4.36490587423552886035306655721, 4.87380771512045236080038238066, 6.19950398655799805319114481220, 6.88536360405237823952775505759, 7.40093744349002691279955792953, 8.167631029198573615278976332254

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