Properties

Label 2-2960-1.1-c1-0-46
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  + 3.15·3-s + 5-s + 0.579·7-s + 6.94·9-s + 0.683·11-s − 7.07·13-s + 3.15·15-s + 7.90·17-s − 0.469·19-s + 1.82·21-s + 3.47·23-s + 25-s + 12.4·27-s + 3.47·29-s − 6.32·31-s + 2.15·33-s + 0.579·35-s − 37-s − 22.2·39-s + 0.683·41-s + 11.5·43-s + 6.94·45-s − 5.00·47-s − 6.66·49-s + 24.9·51-s + 4.50·53-s + 0.683·55-s + ⋯
L(s)  = 1  + 1.82·3-s + 0.447·5-s + 0.219·7-s + 2.31·9-s + 0.206·11-s − 1.96·13-s + 0.814·15-s + 1.91·17-s − 0.107·19-s + 0.399·21-s + 0.724·23-s + 0.200·25-s + 2.39·27-s + 0.644·29-s − 1.13·31-s + 0.375·33-s + 0.0980·35-s − 0.164·37-s − 3.57·39-s + 0.106·41-s + 1.76·43-s + 1.03·45-s − 0.729·47-s − 0.951·49-s + 3.49·51-s + 0.619·53-s + 0.0921·55-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

\(L(1)\) \(\approx\) \(4.181161999\)
\(L(\frac12)\) \(\approx\) \(4.181161999\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.15T + 3T^{2} \)
7 \( 1 - 0.579T + 7T^{2} \)
11 \( 1 - 0.683T + 11T^{2} \)
13 \( 1 + 7.07T + 13T^{2} \)
17 \( 1 - 7.90T + 17T^{2} \)
19 \( 1 + 0.469T + 19T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 - 3.47T + 29T^{2} \)
31 \( 1 + 6.32T + 31T^{2} \)
41 \( 1 - 0.683T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + 5.00T + 47T^{2} \)
53 \( 1 - 4.50T + 53T^{2} \)
59 \( 1 - 6.85T + 59T^{2} \)
61 \( 1 - 3.71T + 61T^{2} \)
67 \( 1 + 7.53T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 - 8.76T + 79T^{2} \)
83 \( 1 + 1.84T + 83T^{2} \)
89 \( 1 + 4.74T + 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−8.812479278633952598504246792973, −7.914059469543316387388135072744, −7.49379121257081172112655252263, −6.85481849318637162185601659344, −5.51175996999113308974156431969, −4.76859581132167961966207955843, −3.77686783366891145878287078657, −2.92579910871870778824239966595, −2.32269191579680501058468020894, −1.28843432005595200464234226723, 1.28843432005595200464234226723, 2.32269191579680501058468020894, 2.92579910871870778824239966595, 3.77686783366891145878287078657, 4.76859581132167961966207955843, 5.51175996999113308974156431969, 6.85481849318637162185601659344, 7.49379121257081172112655252263, 7.914059469543316387388135072744, 8.812479278633952598504246792973

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