Properties

Label 2-2960-1.1-c1-0-13
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  − 2.65·3-s − 5-s − 0.0991·7-s + 4.04·9-s + 0.220·11-s + 4.84·13-s + 2.65·15-s + 5.26·17-s + 5.23·19-s + 0.263·21-s − 7.30·23-s + 25-s − 2.77·27-s − 9.42·29-s − 7.40·31-s − 0.584·33-s + 0.0991·35-s + 37-s − 12.8·39-s − 1.28·41-s + 7.72·43-s − 4.04·45-s − 10.9·47-s − 6.99·49-s − 13.9·51-s + 11.5·53-s − 0.220·55-s + ⋯
L(s)  = 1  − 1.53·3-s − 0.447·5-s − 0.0374·7-s + 1.34·9-s + 0.0663·11-s + 1.34·13-s + 0.685·15-s + 1.27·17-s + 1.20·19-s + 0.0574·21-s − 1.52·23-s + 0.200·25-s − 0.534·27-s − 1.74·29-s − 1.33·31-s − 0.101·33-s + 0.0167·35-s + 0.164·37-s − 2.06·39-s − 0.200·41-s + 1.17·43-s − 0.603·45-s − 1.59·47-s − 0.998·49-s − 1.95·51-s + 1.58·53-s − 0.0296·55-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

\(L(1)\) \(\approx\) \(0.9066347562\)
\(L(\frac12)\) \(\approx\) \(0.9066347562\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.65T + 3T^{2} \)
7 \( 1 + 0.0991T + 7T^{2} \)
11 \( 1 - 0.220T + 11T^{2} \)
13 \( 1 - 4.84T + 13T^{2} \)
17 \( 1 - 5.26T + 17T^{2} \)
19 \( 1 - 5.23T + 19T^{2} \)
23 \( 1 + 7.30T + 23T^{2} \)
29 \( 1 + 9.42T + 29T^{2} \)
31 \( 1 + 7.40T + 31T^{2} \)
41 \( 1 + 1.28T + 41T^{2} \)
43 \( 1 - 7.72T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 - 9.52T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + 3.23T + 67T^{2} \)
71 \( 1 + 8.58T + 71T^{2} \)
73 \( 1 + 3.89T + 73T^{2} \)
79 \( 1 - 3.42T + 79T^{2} \)
83 \( 1 - 1.92T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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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

−8.724331451508675034539325129853, −7.76374698254412804960104295143, −7.24703602885514676753550125570, −6.21938011272280842614043433706, −5.69229626379983502329325258909, −5.18092145623320728791619665511, −3.95619431137689377308849299583, −3.46991858483014668198627786501, −1.69024800834268540727515791017, −0.65647293201216876728097182595, 0.65647293201216876728097182595, 1.69024800834268540727515791017, 3.46991858483014668198627786501, 3.95619431137689377308849299583, 5.18092145623320728791619665511, 5.69229626379983502329325258909, 6.21938011272280842614043433706, 7.24703602885514676753550125570, 7.76374698254412804960104295143, 8.724331451508675034539325129853

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