Properties

Label 2-2960-1.1-c1-0-68
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.76·3-s + 5-s − 1.76·7-s + 0.103·9-s − 4.62·11-s + 1.76·15-s + 2.38·19-s − 3.10·21-s − 4·23-s + 25-s − 5.10·27-s − 7.25·29-s − 1.13·31-s − 8.14·33-s − 1.76·35-s + 37-s + 0.896·41-s − 9.25·43-s + 0.103·45-s + 8.74·47-s − 3.89·49-s + 8.35·53-s − 4.62·55-s + 4.20·57-s + 8.11·59-s + 5.04·61-s − 0.181·63-s + ⋯
L(s)  = 1  + 1.01·3-s + 0.447·5-s − 0.665·7-s + 0.0343·9-s − 1.39·11-s + 0.454·15-s + 0.547·19-s − 0.677·21-s − 0.834·23-s + 0.200·25-s − 0.982·27-s − 1.34·29-s − 0.203·31-s − 1.41·33-s − 0.297·35-s + 0.164·37-s + 0.140·41-s − 1.41·43-s + 0.0153·45-s + 1.27·47-s − 0.556·49-s + 1.14·53-s − 0.623·55-s + 0.557·57-s + 1.05·59-s + 0.646·61-s − 0.0228·63-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: \(1\)
Selberg data: \((2,\ 2960,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.76T + 3T^{2} \)
7 \( 1 + 1.76T + 7T^{2} \)
11 \( 1 + 4.62T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2.38T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 7.25T + 29T^{2} \)
31 \( 1 + 1.13T + 31T^{2} \)
41 \( 1 - 0.896T + 41T^{2} \)
43 \( 1 + 9.25T + 43T^{2} \)
47 \( 1 - 8.74T + 47T^{2} \)
53 \( 1 - 8.35T + 53T^{2} \)
59 \( 1 - 8.11T + 59T^{2} \)
61 \( 1 - 5.04T + 61T^{2} \)
67 \( 1 + 16.1T + 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + 18.0T + 83T^{2} \)
89 \( 1 - 1.45T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−8.495875054585360795802095945980, −7.61704325784401605453653018508, −7.14180844127611195199551233174, −5.84757474444869880476057354559, −5.53109873355808720688838349320, −4.28305067836020154265702049463, −3.29075223151075227607716128858, −2.70432794332268591438512901533, −1.83530597956227026303118743508, 0, 1.83530597956227026303118743508, 2.70432794332268591438512901533, 3.29075223151075227607716128858, 4.28305067836020154265702049463, 5.53109873355808720688838349320, 5.84757474444869880476057354559, 7.14180844127611195199551233174, 7.61704325784401605453653018508, 8.495875054585360795802095945980

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