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Properties

Label 2-637-1.1-c1-0-22
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  + 1.83·2-s + 0.853·3-s + 1.35·4-s + 2.62·5-s + 1.56·6-s − 1.17·8-s − 2.27·9-s + 4.81·10-s + 3.26·11-s + 1.15·12-s + 13-s + 2.24·15-s − 4.87·16-s + 4.53·17-s − 4.16·18-s + 4.06·19-s + 3.56·20-s + 5.98·22-s − 4.53·23-s − 1.00·24-s + 1.89·25-s + 1.83·26-s − 4.50·27-s − 1.42·29-s + 4.10·30-s − 2.80·31-s − 6.57·32-s + ⋯
L(s)  = 1  + 1.29·2-s + 0.492·3-s + 0.678·4-s + 1.17·5-s + 0.638·6-s − 0.416·8-s − 0.757·9-s + 1.52·10-s + 0.984·11-s + 0.334·12-s + 0.277·13-s + 0.578·15-s − 1.21·16-s + 1.09·17-s − 0.980·18-s + 0.932·19-s + 0.797·20-s + 1.27·22-s − 0.945·23-s − 0.205·24-s + 0.378·25-s + 0.359·26-s − 0.866·27-s − 0.264·29-s + 0.749·30-s − 0.503·31-s − 1.16·32-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 3.6585197293.658519729
L(12)L(\frac12) ≈\approx 3.6585197293.658519729
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 1−1.83T+2T2 1 - 1.83T + 2T^{2}
3 1−0.853T+3T2 1 - 0.853T + 3T^{2}
5 1−2.62T+5T2 1 - 2.62T + 5T^{2}
11 1−3.26T+11T2 1 - 3.26T + 11T^{2}
17 1−4.53T+17T2 1 - 4.53T + 17T^{2}
19 1−4.06T+19T2 1 - 4.06T + 19T^{2}
23 1+4.53T+23T2 1 + 4.53T + 23T^{2}
29 1+1.42T+29T2 1 + 1.42T + 29T^{2}
31 1+2.80T+31T2 1 + 2.80T + 31T^{2}
37 1+10.0T+37T2 1 + 10.0T + 37T^{2}
41 1+2.84T+41T2 1 + 2.84T + 41T^{2}
43 1−9.72T+43T2 1 - 9.72T + 43T^{2}
47 1+9.44T+47T2 1 + 9.44T + 47T^{2}
53 1−5.26T+53T2 1 - 5.26T + 53T^{2}
59 1+2.56T+59T2 1 + 2.56T + 59T^{2}
61 1+11.1T+61T2 1 + 11.1T + 61T^{2}
67 1+1.98T+67T2 1 + 1.98T + 67T^{2}
71 1+11.7T+71T2 1 + 11.7T + 71T^{2}
73 1−12.1T+73T2 1 - 12.1T + 73T^{2}
79 1−11.9T+79T2 1 - 11.9T + 79T^{2}
83 1−13.2T+83T2 1 - 13.2T + 83T^{2}
89 1−10.6T+89T2 1 - 10.6T + 89T^{2}
97 1+13.7T+97T2 1 + 13.7T + 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

−10.66354676703402963636632321490, −9.486718033023664631628693345734, −9.120832731181008022570243555899, −7.910692728784159615642134206833, −6.59152220820597614388635124341, −5.79198189725718630430871254835, −5.26562698114258209798116913137, −3.83667107048168242498577795091, −3.09389315517282991663458065506, −1.82639530032567334447259075068, 1.82639530032567334447259075068, 3.09389315517282991663458065506, 3.83667107048168242498577795091, 5.26562698114258209798116913137, 5.79198189725718630430871254835, 6.59152220820597614388635124341, 7.910692728784159615642134206833, 9.120832731181008022570243555899, 9.486718033023664631628693345734, 10.66354676703402963636632321490

Graph of the ZZ-function along the critical line