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Properties

Label 2-2960-1.1-c1-0-24
Degree 22
Conductor 29602960
Sign 11
Analytic cond. 23.635723.6357
Root an. cond. 4.861654.86165
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

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

Λ(s)=(2960s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2960s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29602960    =    24⋅5⋅372^{4} \cdot 5 \cdot 37
Sign: 11
Analytic conductor: 23.635723.6357
Root analytic conductor: 4.861654.86165
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2960, ( :1/2), 1)(2,\ 2960,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4273403032.427340303
L(12)L(\frac12) ≈\approx 2.4273403032.427340303
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1−T 1 - T
37 1+T 1 + T
good3 1−1.59T+3T2 1 - 1.59T + 3T^{2}
7 1+4.50T+7T2 1 + 4.50T + 7T^{2}
11 1−5.01T+11T2 1 - 5.01T + 11T^{2}
13 1+0.0423T+13T2 1 + 0.0423T + 13T^{2}
17 1+3.74T+17T2 1 + 3.74T + 17T^{2}
19 1−5.42T+19T2 1 - 5.42T + 19T^{2}
23 1−4.97T+23T2 1 - 4.97T + 23T^{2}
29 1−4.97T+29T2 1 - 4.97T + 29T^{2}
31 1−4.62T+31T2 1 - 4.62T + 31T^{2}
41 1−5.01T+41T2 1 - 5.01T + 41T^{2}
43 1+0.112T+43T2 1 + 0.112T + 43T^{2}
47 1−1.60T+47T2 1 - 1.60T + 47T^{2}
53 1+10.8T+53T2 1 + 10.8T + 53T^{2}
59 1−9.19T+59T2 1 - 9.19T + 59T^{2}
61 1+8.80T+61T2 1 + 8.80T + 61T^{2}
67 1+0.203T+67T2 1 + 0.203T + 67T^{2}
71 1−13.3T+71T2 1 - 13.3T + 71T^{2}
73 1−7.45T+73T2 1 - 7.45T + 73T^{2}
79 1+6.13T+79T2 1 + 6.13T + 79T^{2}
83 1−17.6T+83T2 1 - 17.6T + 83T^{2}
89 1−17.1T+89T2 1 - 17.1T + 89T^{2}
97 1+4.13T+97T2 1 + 4.13T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line