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Properties

Label 2-3240-1.1-c1-0-12
Degree 22
Conductor 32403240
Sign 11
Analytic cond. 25.871525.8715
Root an. cond. 5.086405.08640
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  + 5-s − 1.44·7-s + 2·17-s + 2.89·19-s + 2.55·23-s + 25-s − 7.89·29-s + 10.8·31-s − 1.44·35-s − 6·37-s + 0.101·41-s − 7.79·43-s + 4.55·47-s − 4.89·49-s + 11.7·53-s + 10.8·59-s − 3·61-s + 11.2·67-s + 9.79·71-s − 5.79·73-s + 2.89·79-s + 0.550·83-s + 2·85-s + 16.7·89-s + 2.89·95-s + 2·97-s + 2·101-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.547·7-s + 0.485·17-s + 0.665·19-s + 0.531·23-s + 0.200·25-s − 1.46·29-s + 1.95·31-s − 0.245·35-s − 0.986·37-s + 0.0157·41-s − 1.18·43-s + 0.663·47-s − 0.699·49-s + 1.62·53-s + 1.41·59-s − 0.384·61-s + 1.37·67-s + 1.16·71-s − 0.678·73-s + 0.326·79-s + 0.0604·83-s + 0.216·85-s + 1.78·89-s + 0.297·95-s + 0.203·97-s + 0.199·101-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 11
Analytic conductor: 25.871525.8715
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3240, ( :1/2), 1)(2,\ 3240,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9188436481.918843648
L(12)L(\frac12) ≈\approx 1.9188436481.918843648
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
good7 1+1.44T+7T2 1 + 1.44T + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1−2.89T+19T2 1 - 2.89T + 19T^{2}
23 1−2.55T+23T2 1 - 2.55T + 23T^{2}
29 1+7.89T+29T2 1 + 7.89T + 29T^{2}
31 1−10.8T+31T2 1 - 10.8T + 31T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1−0.101T+41T2 1 - 0.101T + 41T^{2}
43 1+7.79T+43T2 1 + 7.79T + 43T^{2}
47 1−4.55T+47T2 1 - 4.55T + 47T^{2}
53 1−11.7T+53T2 1 - 11.7T + 53T^{2}
59 1−10.8T+59T2 1 - 10.8T + 59T^{2}
61 1+3T+61T2 1 + 3T + 61T^{2}
67 1−11.2T+67T2 1 - 11.2T + 67T^{2}
71 1−9.79T+71T2 1 - 9.79T + 71T^{2}
73 1+5.79T+73T2 1 + 5.79T + 73T^{2}
79 1−2.89T+79T2 1 - 2.89T + 79T^{2}
83 1−0.550T+83T2 1 - 0.550T + 83T^{2}
89 1−16.7T+89T2 1 - 16.7T + 89T^{2}
97 1−2T+97T2 1 - 2T + 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

−8.708854249634062873437283848740, −7.923685810313280565397389556111, −7.05642717890901162506008104396, −6.46727990629738871001075173803, −5.56554023586073573580900961129, −4.99341645588020936344215002534, −3.81827485177166097005396021959, −3.08777468354378049008845670357, −2.07683890516739483340949947699, −0.841177527231864577721067200852, 0.841177527231864577721067200852, 2.07683890516739483340949947699, 3.08777468354378049008845670357, 3.81827485177166097005396021959, 4.99341645588020936344215002534, 5.56554023586073573580900961129, 6.46727990629738871001075173803, 7.05642717890901162506008104396, 7.923685810313280565397389556111, 8.708854249634062873437283848740

Graph of the ZZ-function along the critical line