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Properties

Label 2-637-1.1-c1-0-0
Degree 22
Conductor 637637
Sign 11
Analytic cond. 5.086475.08647
Root an. cond. 2.255322.25532
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  − 2.00·2-s − 1.75·3-s + 2.03·4-s − 0.905·5-s + 3.53·6-s − 0.0686·8-s + 0.0942·9-s + 1.81·10-s + 0.716·11-s − 3.57·12-s − 13-s + 1.59·15-s − 3.93·16-s − 2.35·17-s − 0.189·18-s − 6.63·19-s − 1.84·20-s − 1.43·22-s + 3.75·23-s + 0.120·24-s − 4.17·25-s + 2.00·26-s + 5.11·27-s + 3.25·29-s − 3.20·30-s − 1.57·31-s + 8.03·32-s + ⋯
L(s)  = 1  − 1.42·2-s − 1.01·3-s + 1.01·4-s − 0.405·5-s + 1.44·6-s − 0.0242·8-s + 0.0314·9-s + 0.575·10-s + 0.215·11-s − 1.03·12-s − 0.277·13-s + 0.411·15-s − 0.982·16-s − 0.570·17-s − 0.0446·18-s − 1.52·19-s − 0.411·20-s − 0.306·22-s + 0.783·23-s + 0.0246·24-s − 0.835·25-s + 0.393·26-s + 0.983·27-s + 0.604·29-s − 0.584·30-s − 0.282·31-s + 1.41·32-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 11
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 1)(2,\ 637,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.30606074500.3060607450
L(12)L(\frac12) ≈\approx 0.30606074500.3060607450
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)
bad7 1 1
13 1+T 1 + T
good2 1+2.00T+2T2 1 + 2.00T + 2T^{2}
3 1+1.75T+3T2 1 + 1.75T + 3T^{2}
5 1+0.905T+5T2 1 + 0.905T + 5T^{2}
11 1−0.716T+11T2 1 - 0.716T + 11T^{2}
17 1+2.35T+17T2 1 + 2.35T + 17T^{2}
19 1+6.63T+19T2 1 + 6.63T + 19T^{2}
23 1−3.75T+23T2 1 - 3.75T + 23T^{2}
29 1−3.25T+29T2 1 - 3.25T + 29T^{2}
31 1+1.57T+31T2 1 + 1.57T + 31T^{2}
37 1−5.20T+37T2 1 - 5.20T + 37T^{2}
41 1+4.92T+41T2 1 + 4.92T + 41T^{2}
43 1+9.43T+43T2 1 + 9.43T + 43T^{2}
47 1−8.31T+47T2 1 - 8.31T + 47T^{2}
53 1−14.0T+53T2 1 - 14.0T + 53T^{2}
59 1+0.716T+59T2 1 + 0.716T + 59T^{2}
61 1−11.6T+61T2 1 - 11.6T + 61T^{2}
67 1−9.39T+67T2 1 - 9.39T + 67T^{2}
71 1−10.9T+71T2 1 - 10.9T + 71T^{2}
73 1−3.47T+73T2 1 - 3.47T + 73T^{2}
79 1−13.0T+79T2 1 - 13.0T + 79T^{2}
83 1+3.54T+83T2 1 + 3.54T + 83T^{2}
89 1+12.0T+89T2 1 + 12.0T + 89T^{2}
97 1+7.43T+97T2 1 + 7.43T + 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

−10.57323558578137676195984937419, −9.793600687529902077373433257744, −8.738217503189565302578205662201, −8.227241499656225880955860428158, −7.03542766398972229338840233414, −6.45151260537239723045933174400, −5.19146952663274279601492832562, −4.11950990912236128083857857925, −2.25130907375641470428688420174, −0.59997345471379059232210335297, 0.59997345471379059232210335297, 2.25130907375641470428688420174, 4.11950990912236128083857857925, 5.19146952663274279601492832562, 6.45151260537239723045933174400, 7.03542766398972229338840233414, 8.227241499656225880955860428158, 8.738217503189565302578205662201, 9.793600687529902077373433257744, 10.57323558578137676195984937419

Graph of the ZZ-function along the critical line