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Properties

Label 2-637-1.1-c1-0-11
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  − 0.195·2-s − 0.259·3-s − 1.96·4-s + 3.93·5-s + 0.0508·6-s + 0.775·8-s − 2.93·9-s − 0.769·10-s + 4.50·11-s + 0.509·12-s + 13-s − 1.02·15-s + 3.77·16-s − 2.28·17-s + 0.573·18-s − 1.78·19-s − 7.71·20-s − 0.881·22-s + 1.74·23-s − 0.201·24-s + 10.4·25-s − 0.195·26-s + 1.54·27-s + 1.65·29-s + 0.199·30-s + 5.60·31-s − 2.28·32-s + ⋯
L(s)  = 1  − 0.138·2-s − 0.149·3-s − 0.980·4-s + 1.75·5-s + 0.0207·6-s + 0.274·8-s − 0.977·9-s − 0.243·10-s + 1.35·11-s + 0.147·12-s + 0.277·13-s − 0.263·15-s + 0.942·16-s − 0.553·17-s + 0.135·18-s − 0.410·19-s − 1.72·20-s − 0.187·22-s + 0.362·23-s − 0.0411·24-s + 2.09·25-s − 0.0383·26-s + 0.296·27-s + 0.306·29-s + 0.0364·30-s + 1.00·31-s − 0.404·32-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 1.4579315171.457931517
L(12)L(\frac12) ≈\approx 1.4579315171.457931517
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+0.195T+2T2 1 + 0.195T + 2T^{2}
3 1+0.259T+3T2 1 + 0.259T + 3T^{2}
5 1−3.93T+5T2 1 - 3.93T + 5T^{2}
11 1−4.50T+11T2 1 - 4.50T + 11T^{2}
17 1+2.28T+17T2 1 + 2.28T + 17T^{2}
19 1+1.78T+19T2 1 + 1.78T + 19T^{2}
23 1−1.74T+23T2 1 - 1.74T + 23T^{2}
29 1−1.65T+29T2 1 - 1.65T + 29T^{2}
31 1−5.60T+31T2 1 - 5.60T + 31T^{2}
37 1−7.14T+37T2 1 - 7.14T + 37T^{2}
41 1+8.11T+41T2 1 + 8.11T + 41T^{2}
43 1−6.81T+43T2 1 - 6.81T + 43T^{2}
47 1−3.54T+47T2 1 - 3.54T + 47T^{2}
53 1−3.28T+53T2 1 - 3.28T + 53T^{2}
59 1−4.50T+59T2 1 - 4.50T + 59T^{2}
61 1−7.54T+61T2 1 - 7.54T + 61T^{2}
67 1+12.6T+67T2 1 + 12.6T + 67T^{2}
71 1−9.54T+71T2 1 - 9.54T + 71T^{2}
73 1−1.08T+73T2 1 - 1.08T + 73T^{2}
79 1−0.791T+79T2 1 - 0.791T + 79T^{2}
83 1+7.14T+83T2 1 + 7.14T + 83T^{2}
89 1+11.2T+89T2 1 + 11.2T + 89T^{2}
97 1+8.81T+97T2 1 + 8.81T + 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

−10.36110620479067596003155069244, −9.572817839349911366904369790832, −8.975133828205401973565169243250, −8.392307526431928522157372431685, −6.69757501810445103551369381232, −6.02883304303346113051358692948, −5.22509066933486439733252162204, −4.12836263864436883298067059119, −2.63643294720511834793319575028, −1.19271414644531081978401165037, 1.19271414644531081978401165037, 2.63643294720511834793319575028, 4.12836263864436883298067059119, 5.22509066933486439733252162204, 6.02883304303346113051358692948, 6.69757501810445103551369381232, 8.392307526431928522157372431685, 8.975133828205401973565169243250, 9.572817839349911366904369790832, 10.36110620479067596003155069244

Graph of the ZZ-function along the critical line