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Properties

Label 2-605-1.1-c1-0-7
Degree 22
Conductor 605605
Sign 11
Analytic cond. 4.830944.83094
Root an. cond. 2.197942.19794
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  − 1.87·2-s − 1.77·3-s + 1.51·4-s + 5-s + 3.33·6-s + 4.25·7-s + 0.901·8-s + 0.159·9-s − 1.87·10-s − 2.70·12-s + 1.36·13-s − 7.98·14-s − 1.77·15-s − 4.73·16-s + 2.09·17-s − 0.299·18-s + 0.604·19-s + 1.51·20-s − 7.56·21-s − 4.39·23-s − 1.60·24-s + 25-s − 2.55·26-s + 5.04·27-s + 6.46·28-s − 6.63·29-s + 3.33·30-s + ⋯
L(s)  = 1  − 1.32·2-s − 1.02·3-s + 0.759·4-s + 0.447·5-s + 1.36·6-s + 1.60·7-s + 0.318·8-s + 0.0531·9-s − 0.593·10-s − 0.779·12-s + 0.377·13-s − 2.13·14-s − 0.458·15-s − 1.18·16-s + 0.508·17-s − 0.0705·18-s + 0.138·19-s + 0.339·20-s − 1.65·21-s − 0.916·23-s − 0.327·24-s + 0.200·25-s − 0.500·26-s + 0.971·27-s + 1.22·28-s − 1.23·29-s + 0.608·30-s + ⋯

Functional equation

Λ(s)=(605s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(605s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 605605    =    5⋅1125 \cdot 11^{2}
Sign: 11
Analytic conductor: 4.830944.83094
Root analytic conductor: 2.197942.19794
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 605, ( :1/2), 1)(2,\ 605,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.64474200720.6447420072
L(12)L(\frac12) ≈\approx 0.64474200720.6447420072
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)
bad5 1−T 1 - T
11 1 1
good2 1+1.87T+2T2 1 + 1.87T + 2T^{2}
3 1+1.77T+3T2 1 + 1.77T + 3T^{2}
7 1−4.25T+7T2 1 - 4.25T + 7T^{2}
13 1−1.36T+13T2 1 - 1.36T + 13T^{2}
17 1−2.09T+17T2 1 - 2.09T + 17T^{2}
19 1−0.604T+19T2 1 - 0.604T + 19T^{2}
23 1+4.39T+23T2 1 + 4.39T + 23T^{2}
29 1+6.63T+29T2 1 + 6.63T + 29T^{2}
31 1+2.19T+31T2 1 + 2.19T + 31T^{2}
37 1−6.16T+37T2 1 - 6.16T + 37T^{2}
41 1−7.40T+41T2 1 - 7.40T + 41T^{2}
43 1−12.6T+43T2 1 - 12.6T + 43T^{2}
47 1+3.07T+47T2 1 + 3.07T + 47T^{2}
53 1+6.65T+53T2 1 + 6.65T + 53T^{2}
59 1−12.0T+59T2 1 - 12.0T + 59T^{2}
61 1+5.68T+61T2 1 + 5.68T + 61T^{2}
67 1−9.86T+67T2 1 - 9.86T + 67T^{2}
71 1+5.23T+71T2 1 + 5.23T + 71T^{2}
73 1+0.722T+73T2 1 + 0.722T + 73T^{2}
79 1−5.67T+79T2 1 - 5.67T + 79T^{2}
83 1+0.952T+83T2 1 + 0.952T + 83T^{2}
89 1−1.24T+89T2 1 - 1.24T + 89T^{2}
97 1−11.5T+97T2 1 - 11.5T + 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

−10.81923980747304794773415624293, −9.789450759579999885227731640820, −8.941778165051189525738264100272, −8.030009234004533529306819749843, −7.44807360774094599999828725945, −6.10107983838590632936608903311, −5.33322061232929930404373761811, −4.30527704672202159909961329509, −2.08303021296616042965844033710, −0.948431305586601174813472817241, 0.948431305586601174813472817241, 2.08303021296616042965844033710, 4.30527704672202159909961329509, 5.33322061232929930404373761811, 6.10107983838590632936608903311, 7.44807360774094599999828725945, 8.030009234004533529306819749843, 8.941778165051189525738264100272, 9.789450759579999885227731640820, 10.81923980747304794773415624293

Graph of the ZZ-function along the critical line