Properties

Label 2-2960-1.1-c1-0-39
Degree $2$
Conductor $2960$
Sign $1$
Analytic cond. $23.6357$
Root an. cond. $4.86165$
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  + 1.77·3-s + 5-s + 2.66·7-s + 0.148·9-s + 2.35·11-s + 0.522·13-s + 1.77·15-s + 5.20·17-s + 5.03·19-s + 4.73·21-s + 2.63·23-s + 25-s − 5.05·27-s + 0.177·29-s − 6.99·31-s + 4.17·33-s + 2.66·35-s − 37-s + 0.927·39-s − 4.87·41-s − 12.0·43-s + 0.148·45-s − 1.98·47-s + 0.124·49-s + 9.23·51-s + 3.19·53-s + 2.35·55-s + ⋯
L(s)  = 1  + 1.02·3-s + 0.447·5-s + 1.00·7-s + 0.0495·9-s + 0.709·11-s + 0.144·13-s + 0.458·15-s + 1.26·17-s + 1.15·19-s + 1.03·21-s + 0.548·23-s + 0.200·25-s − 0.973·27-s + 0.0330·29-s − 1.25·31-s + 0.726·33-s + 0.451·35-s − 0.164·37-s + 0.148·39-s − 0.760·41-s − 1.83·43-s + 0.0221·45-s − 0.289·47-s + 0.0178·49-s + 1.29·51-s + 0.439·53-s + 0.317·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(23.6357\)
Root analytic conductor: \(4.86165\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.554856150\)
\(L(\frac12)\) \(\approx\) \(3.554856150\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
37 \( 1 + T \)
good3 \( 1 - 1.77T + 3T^{2} \)
7 \( 1 - 2.66T + 7T^{2} \)
11 \( 1 - 2.35T + 11T^{2} \)
13 \( 1 - 0.522T + 13T^{2} \)
17 \( 1 - 5.20T + 17T^{2} \)
19 \( 1 - 5.03T + 19T^{2} \)
23 \( 1 - 2.63T + 23T^{2} \)
29 \( 1 - 0.177T + 29T^{2} \)
31 \( 1 + 6.99T + 31T^{2} \)
41 \( 1 + 4.87T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + 1.98T + 47T^{2} \)
53 \( 1 - 3.19T + 53T^{2} \)
59 \( 1 + 1.24T + 59T^{2} \)
61 \( 1 - 3.72T + 61T^{2} \)
67 \( 1 - 4.11T + 67T^{2} \)
71 \( 1 + 4.17T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 + 0.201T + 79T^{2} \)
83 \( 1 - 6.51T + 83T^{2} \)
89 \( 1 - 7.87T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−8.709094197169856061812893284098, −8.063281060846413612356462676443, −7.47208637373524309400670910628, −6.58844272939668861806368620119, −5.46536403956889888055530070172, −5.03175813916357883387821787601, −3.69567162456136255185510272759, −3.20352268822664926872059903887, −2.01553270786878698410248675393, −1.25354886023109796862706243233, 1.25354886023109796862706243233, 2.01553270786878698410248675393, 3.20352268822664926872059903887, 3.69567162456136255185510272759, 5.03175813916357883387821787601, 5.46536403956889888055530070172, 6.58844272939668861806368620119, 7.47208637373524309400670910628, 8.063281060846413612356462676443, 8.709094197169856061812893284098

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