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Properties

Label 2-3240-1.1-c1-0-19
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 + 3.37·7-s + 2.37·11-s − 3.37·13-s − 6.74·17-s + 19-s + 5.37·23-s + 25-s + 2.37·29-s + 11.1·31-s + 3.37·35-s + 6·37-s + 0.255·41-s − 4.74·43-s − 9.37·47-s + 4.37·49-s + 10.1·53-s + 2.37·55-s + 5·59-s + 12.7·61-s − 3.37·65-s − 0.744·67-s − 4.37·71-s + 14.7·73-s + 8·77-s − 2.74·79-s − 10·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.27·7-s + 0.715·11-s − 0.935·13-s − 1.63·17-s + 0.229·19-s + 1.12·23-s + 0.200·25-s + 0.440·29-s + 1.99·31-s + 0.570·35-s + 0.986·37-s + 0.0398·41-s − 0.723·43-s − 1.36·47-s + 0.624·49-s + 1.38·53-s + 0.319·55-s + 0.650·59-s + 1.63·61-s − 0.418·65-s − 0.0909·67-s − 0.518·71-s + 1.72·73-s + 0.911·77-s − 0.308·79-s − 1.09·83-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 2.4425323652.442532365
L(12)L(\frac12) ≈\approx 2.4425323652.442532365
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 1−3.37T+7T2 1 - 3.37T + 7T^{2}
11 1−2.37T+11T2 1 - 2.37T + 11T^{2}
13 1+3.37T+13T2 1 + 3.37T + 13T^{2}
17 1+6.74T+17T2 1 + 6.74T + 17T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1−5.37T+23T2 1 - 5.37T + 23T^{2}
29 1−2.37T+29T2 1 - 2.37T + 29T^{2}
31 1−11.1T+31T2 1 - 11.1T + 31T^{2}
37 1−6T+37T2 1 - 6T + 37T^{2}
41 1−0.255T+41T2 1 - 0.255T + 41T^{2}
43 1+4.74T+43T2 1 + 4.74T + 43T^{2}
47 1+9.37T+47T2 1 + 9.37T + 47T^{2}
53 1−10.1T+53T2 1 - 10.1T + 53T^{2}
59 1−5T+59T2 1 - 5T + 59T^{2}
61 1−12.7T+61T2 1 - 12.7T + 61T^{2}
67 1+0.744T+67T2 1 + 0.744T + 67T^{2}
71 1+4.37T+71T2 1 + 4.37T + 71T^{2}
73 1−14.7T+73T2 1 - 14.7T + 73T^{2}
79 1+2.74T+79T2 1 + 2.74T + 79T^{2}
83 1+10T+83T2 1 + 10T + 83T^{2}
89 1+4.37T+89T2 1 + 4.37T + 89T^{2}
97 1+4.74T+97T2 1 + 4.74T + 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

−8.519795717253944912663599767676, −8.127776229435482206257163639049, −6.93233445152459762775167582449, −6.65707135671478672315207342489, −5.46860093726808454451869375991, −4.73846595381922484920980904226, −4.27180001033612376583202349250, −2.82634096317416370562432484461, −2.05625676713037649228336489776, −0.987532938810142313325817271311, 0.987532938810142313325817271311, 2.05625676713037649228336489776, 2.82634096317416370562432484461, 4.27180001033612376583202349250, 4.73846595381922484920980904226, 5.46860093726808454451869375991, 6.65707135671478672315207342489, 6.93233445152459762775167582449, 8.127776229435482206257163639049, 8.519795717253944912663599767676

Graph of the ZZ-function along the critical line