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Properties

Label 2-475-1.1-c1-0-22
Degree 22
Conductor 475475
Sign 11
Analytic cond. 3.792893.79289
Root an. cond. 1.947531.94753
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  + 2.80·2-s + 0.554·3-s + 5.85·4-s + 1.55·6-s − 3.04·7-s + 10.7·8-s − 2.69·9-s − 2.93·11-s + 3.24·12-s + 3.24·13-s − 8.54·14-s + 18.5·16-s − 2.15·17-s − 7.54·18-s − 19-s − 1.69·21-s − 8.23·22-s + 1.19·23-s + 5.98·24-s + 9.09·26-s − 3.15·27-s − 17.8·28-s − 1.77·29-s − 9.34·31-s + 30.3·32-s − 1.63·33-s − 6.04·34-s + ⋯
L(s)  = 1  + 1.98·2-s + 0.320·3-s + 2.92·4-s + 0.634·6-s − 1.15·7-s + 3.81·8-s − 0.897·9-s − 0.886·11-s + 0.937·12-s + 0.900·13-s − 2.28·14-s + 4.63·16-s − 0.523·17-s − 1.77·18-s − 0.229·19-s − 0.369·21-s − 1.75·22-s + 0.249·23-s + 1.22·24-s + 1.78·26-s − 0.607·27-s − 3.37·28-s − 0.329·29-s − 1.67·31-s + 5.36·32-s − 0.283·33-s − 1.03·34-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 11
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 475, ( :1/2), 1)(2,\ 475,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.3522598654.352259865
L(12)L(\frac12) ≈\approx 4.3522598654.352259865
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 1
19 1+T 1 + T
good2 1−2.80T+2T2 1 - 2.80T + 2T^{2}
3 1−0.554T+3T2 1 - 0.554T + 3T^{2}
7 1+3.04T+7T2 1 + 3.04T + 7T^{2}
11 1+2.93T+11T2 1 + 2.93T + 11T^{2}
13 1−3.24T+13T2 1 - 3.24T + 13T^{2}
17 1+2.15T+17T2 1 + 2.15T + 17T^{2}
23 1−1.19T+23T2 1 - 1.19T + 23T^{2}
29 1+1.77T+29T2 1 + 1.77T + 29T^{2}
31 1+9.34T+31T2 1 + 9.34T + 31T^{2}
37 1+1.15T+37T2 1 + 1.15T + 37T^{2}
41 1−8.57T+41T2 1 - 8.57T + 41T^{2}
43 1−5.27T+43T2 1 - 5.27T + 43T^{2}
47 1−2.35T+47T2 1 - 2.35T + 47T^{2}
53 1−8.82T+53T2 1 - 8.82T + 53T^{2}
59 1+5.70T+59T2 1 + 5.70T + 59T^{2}
61 1+9.96T+61T2 1 + 9.96T + 61T^{2}
67 1−4.98T+67T2 1 - 4.98T + 67T^{2}
71 1−2.70T+71T2 1 - 2.70T + 71T^{2}
73 1−13.7T+73T2 1 - 13.7T + 73T^{2}
79 1−5.66T+79T2 1 - 5.66T + 79T^{2}
83 1+3.00T+83T2 1 + 3.00T + 83T^{2}
89 1+10.2T+89T2 1 + 10.2T + 89T^{2}
97 1−3.24T+97T2 1 - 3.24T + 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

−11.03986268066041007923678197459, −10.76096453834377468072192542832, −9.275746539685960745523985903126, −7.941645861540312886436099168519, −6.93339324115327041452993030567, −6.00895571247158549336326661072, −5.42086286315542444767795729610, −4.03547890334950797372237490482, −3.21317720562659336125363041437, −2.33701947189199226304313544048, 2.33701947189199226304313544048, 3.21317720562659336125363041437, 4.03547890334950797372237490482, 5.42086286315542444767795729610, 6.00895571247158549336326661072, 6.93339324115327041452993030567, 7.941645861540312886436099168519, 9.275746539685960745523985903126, 10.76096453834377468072192542832, 11.03986268066041007923678197459

Graph of the ZZ-function along the critical line