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Properties

Label 2-627-1.1-c1-0-2
Degree 22
Conductor 627627
Sign 11
Analytic cond. 5.006625.00662
Root an. cond. 2.237542.23754
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  − 0.696·2-s + 3-s − 1.51·4-s − 2.51·5-s − 0.696·6-s − 2.32·7-s + 2.44·8-s + 9-s + 1.75·10-s − 11-s − 1.51·12-s + 1.69·13-s + 1.61·14-s − 2.51·15-s + 1.32·16-s + 7.58·17-s − 0.696·18-s − 19-s + 3.81·20-s − 2.32·21-s + 0.696·22-s + 0.248·23-s + 2.44·24-s + 1.32·25-s − 1.18·26-s + 27-s + 3.52·28-s + ⋯
L(s)  = 1  − 0.492·2-s + 0.577·3-s − 0.757·4-s − 1.12·5-s − 0.284·6-s − 0.878·7-s + 0.865·8-s + 0.333·9-s + 0.553·10-s − 0.301·11-s − 0.437·12-s + 0.470·13-s + 0.432·14-s − 0.649·15-s + 0.331·16-s + 1.83·17-s − 0.164·18-s − 0.229·19-s + 0.852·20-s − 0.507·21-s + 0.148·22-s + 0.0518·23-s + 0.499·24-s + 0.265·25-s − 0.231·26-s + 0.192·27-s + 0.665·28-s + ⋯

Functional equation

Λ(s)=(627s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(627s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 627627    =    3⋅11⋅193 \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 5.006625.00662
Root analytic conductor: 2.237542.23754
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 627, ( :1/2), 1)(2,\ 627,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.85378369980.8537836998
L(12)L(\frac12) ≈\approx 0.85378369980.8537836998
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)
bad3 1−T 1 - T
11 1+T 1 + T
19 1+T 1 + T
good2 1+0.696T+2T2 1 + 0.696T + 2T^{2}
5 1+2.51T+5T2 1 + 2.51T + 5T^{2}
7 1+2.32T+7T2 1 + 2.32T + 7T^{2}
13 1−1.69T+13T2 1 - 1.69T + 13T^{2}
17 1−7.58T+17T2 1 - 7.58T + 17T^{2}
23 1−0.248T+23T2 1 - 0.248T + 23T^{2}
29 1−10.2T+29T2 1 - 10.2T + 29T^{2}
31 1−2.07T+31T2 1 - 2.07T + 31T^{2}
37 1−11.4T+37T2 1 - 11.4T + 37T^{2}
41 1−4.59T+41T2 1 - 4.59T + 41T^{2}
43 1+12.6T+43T2 1 + 12.6T + 43T^{2}
47 1−2.84T+47T2 1 - 2.84T + 47T^{2}
53 1+1.65T+53T2 1 + 1.65T + 53T^{2}
59 1−1.06T+59T2 1 - 1.06T + 59T^{2}
61 1−5.30T+61T2 1 - 5.30T + 61T^{2}
67 1−0.303T+67T2 1 - 0.303T + 67T^{2}
71 1−12.4T+71T2 1 - 12.4T + 71T^{2}
73 1+14.2T+73T2 1 + 14.2T + 73T^{2}
79 1+1.81T+79T2 1 + 1.81T + 79T^{2}
83 1−9.16T+83T2 1 - 9.16T + 83T^{2}
89 1+6.59T+89T2 1 + 6.59T + 89T^{2}
97 1−8.72T+97T2 1 - 8.72T + 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

−10.14214238403987079433163187087, −9.855546284893027412325116176316, −8.704974591706763500573945728239, −8.079307424140824278202677101547, −7.49418691152323461639661794478, −6.22428184642417359821541244566, −4.83842175090924746385875659925, −3.83473079939382034289542235976, −3.05576065421609926345454169925, −0.860472693055647410967144995114, 0.860472693055647410967144995114, 3.05576065421609926345454169925, 3.83473079939382034289542235976, 4.83842175090924746385875659925, 6.22428184642417359821541244566, 7.49418691152323461639661794478, 8.079307424140824278202677101547, 8.704974591706763500573945728239, 9.855546284893027412325116176316, 10.14214238403987079433163187087

Graph of the ZZ-function along the critical line