Properties

Label 2-2960-1.1-c1-0-52
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  − 3.12·3-s + 5-s + 3.12·7-s + 6.76·9-s − 1.51·11-s − 3.12·15-s − 5.60·19-s − 9.76·21-s − 4·23-s + 25-s − 11.7·27-s − 1.03·29-s + 0.640·31-s + 4.73·33-s + 3.12·35-s + 37-s − 5.76·41-s − 3.03·43-s + 6.76·45-s + 1.18·47-s + 2.76·49-s + 8.79·53-s − 1.51·55-s + 17.5·57-s + 3.67·59-s − 14.4·61-s + 21.1·63-s + ⋯
L(s)  = 1  − 1.80·3-s + 0.447·5-s + 1.18·7-s + 2.25·9-s − 0.456·11-s − 0.806·15-s − 1.28·19-s − 2.13·21-s − 0.834·23-s + 0.200·25-s − 2.26·27-s − 0.191·29-s + 0.114·31-s + 0.824·33-s + 0.528·35-s + 0.164·37-s − 0.900·41-s − 0.462·43-s + 1.00·45-s + 0.172·47-s + 0.394·49-s + 1.20·53-s − 0.204·55-s + 2.32·57-s + 0.477·59-s − 1.85·61-s + 2.66·63-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: $\chi_{2960} (1, \cdot )$
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 \( 1 + 3.12T + 3T^{2} \)
7 \( 1 - 3.12T + 7T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.60T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 1.03T + 29T^{2} \)
31 \( 1 - 0.640T + 31T^{2} \)
41 \( 1 + 5.76T + 41T^{2} \)
43 \( 1 + 3.03T + 43T^{2} \)
47 \( 1 - 1.18T + 47T^{2} \)
53 \( 1 - 8.79T + 53T^{2} \)
59 \( 1 - 3.67T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 - 0.734T + 71T^{2} \)
73 \( 1 - 7.70T + 73T^{2} \)
79 \( 1 - 2.57T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 - 8.56T + 89T^{2} \)
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

−8.226985393869600505416713418882, −7.52819679329816425181517887959, −6.57797412407734180169563053998, −6.05436070894271714123132871619, −5.24308442386988117172680867809, −4.77704454272368536988252881158, −3.96510771611150243270936538530, −2.21582638801360755608465398356, −1.35540703183763351804611562360, 0, 1.35540703183763351804611562360, 2.21582638801360755608465398356, 3.96510771611150243270936538530, 4.77704454272368536988252881158, 5.24308442386988117172680867809, 6.05436070894271714123132871619, 6.57797412407734180169563053998, 7.52819679329816425181517887959, 8.226985393869600505416713418882

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