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Properties

Label 2-3960-1.1-c1-0-11
Degree 22
Conductor 39603960
Sign 11
Analytic cond. 31.620731.6207
Root an. cond. 5.623235.62323
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s − 11-s + 8·17-s − 8·19-s − 4·23-s + 25-s + 6·29-s − 2·35-s + 6·37-s + 2·41-s + 2·43-s + 4·47-s − 3·49-s + 2·53-s − 55-s + 12·59-s − 6·61-s + 8·67-s − 8·73-s + 2·77-s + 4·79-s + 6·83-s + 8·85-s + 10·89-s − 8·95-s − 10·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.301·11-s + 1.94·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.338·35-s + 0.986·37-s + 0.312·41-s + 0.304·43-s + 0.583·47-s − 3/7·49-s + 0.274·53-s − 0.134·55-s + 1.56·59-s − 0.768·61-s + 0.977·67-s − 0.936·73-s + 0.227·77-s + 0.450·79-s + 0.658·83-s + 0.867·85-s + 1.05·89-s − 0.820·95-s − 1.01·97-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 31.620731.6207
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3960, ( :1/2), 1)(2,\ 3960,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7579329271.757932927
L(12)L(\frac12) ≈\approx 1.7579329271.757932927
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
3 1 1
5 1−T 1 - T
11 1+T 1 + T
good7 1+2T+pT2 1 + 2 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−8T+pT2 1 - 8 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p T^{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

−8.378529573197428734409008106681, −7.83877309088964992232876588117, −6.89297214817942897916236648043, −6.14340315396523346146302421076, −5.70325274306668947557808369172, −4.67076738944322484715966029500, −3.80817728324857729378799634204, −2.92165762439534383333189009412, −2.07609416527182384876181592530, −0.75347698742480715400167322922, 0.75347698742480715400167322922, 2.07609416527182384876181592530, 2.92165762439534383333189009412, 3.80817728324857729378799634204, 4.67076738944322484715966029500, 5.70325274306668947557808369172, 6.14340315396523346146302421076, 6.89297214817942897916236648043, 7.83877309088964992232876588117, 8.378529573197428734409008106681

Graph of the ZZ-function along the critical line