Properties

Label 2-2960-1.1-c1-0-24
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  + 1.59·3-s + 5-s − 4.50·7-s − 0.463·9-s + 5.01·11-s − 0.0423·13-s + 1.59·15-s − 3.74·17-s + 5.42·19-s − 7.17·21-s + 4.97·23-s + 25-s − 5.51·27-s + 4.97·29-s + 4.62·31-s + 7.98·33-s − 4.50·35-s − 37-s − 0.0674·39-s + 5.01·41-s − 0.112·43-s − 0.463·45-s + 1.60·47-s + 13.2·49-s − 5.96·51-s − 10.8·53-s + 5.01·55-s + ⋯
L(s)  = 1  + 0.919·3-s + 0.447·5-s − 1.70·7-s − 0.154·9-s + 1.51·11-s − 0.0117·13-s + 0.411·15-s − 0.908·17-s + 1.24·19-s − 1.56·21-s + 1.03·23-s + 0.200·25-s − 1.06·27-s + 0.923·29-s + 0.830·31-s + 1.39·33-s − 0.761·35-s − 0.164·37-s − 0.0108·39-s + 0.783·41-s − 0.0171·43-s − 0.0690·45-s + 0.234·47-s + 1.89·49-s − 0.835·51-s − 1.49·53-s + 0.676·55-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: $\chi_{2960} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.427340303\)
\(L(\frac12)\) \(\approx\) \(2.427340303\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.59T + 3T^{2} \)
7 \( 1 + 4.50T + 7T^{2} \)
11 \( 1 - 5.01T + 11T^{2} \)
13 \( 1 + 0.0423T + 13T^{2} \)
17 \( 1 + 3.74T + 17T^{2} \)
19 \( 1 - 5.42T + 19T^{2} \)
23 \( 1 - 4.97T + 23T^{2} \)
29 \( 1 - 4.97T + 29T^{2} \)
31 \( 1 - 4.62T + 31T^{2} \)
41 \( 1 - 5.01T + 41T^{2} \)
43 \( 1 + 0.112T + 43T^{2} \)
47 \( 1 - 1.60T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 9.19T + 59T^{2} \)
61 \( 1 + 8.80T + 61T^{2} \)
67 \( 1 + 0.203T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 7.45T + 73T^{2} \)
79 \( 1 + 6.13T + 79T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 - 17.1T + 89T^{2} \)
97 \( 1 + 4.13T + 97T^{2} \)
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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

−9.106306797431896505056875716339, −8.145818871767513569312603451342, −7.06741574233738803360175053058, −6.53804098740504834081467345659, −5.95465162516394125603627833713, −4.76268660268179950266234507624, −3.65036214397222232239846010032, −3.15117425739817709490130486258, −2.33804573798431334728166562072, −0.923568043368975551631184026104, 0.923568043368975551631184026104, 2.33804573798431334728166562072, 3.15117425739817709490130486258, 3.65036214397222232239846010032, 4.76268660268179950266234507624, 5.95465162516394125603627833713, 6.53804098740504834081467345659, 7.06741574233738803360175053058, 8.145818871767513569312603451342, 9.106306797431896505056875716339

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