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Properties

Label 2-627-1.1-c1-0-30
Degree 22
Conductor 627627
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.24·2-s − 3-s + 3.04·4-s − 4.04·5-s − 2.24·6-s − 1.80·7-s + 2.35·8-s + 9-s − 9.09·10-s − 11-s − 3.04·12-s − 4.13·13-s − 4.04·14-s + 4.04·15-s − 0.801·16-s − 3.35·17-s + 2.24·18-s − 19-s − 12.3·20-s + 1.80·21-s − 2.24·22-s + 7.31·23-s − 2.35·24-s + 11.3·25-s − 9.29·26-s − 27-s − 5.49·28-s + ⋯
L(s)  = 1  + 1.58·2-s − 0.577·3-s + 1.52·4-s − 1.81·5-s − 0.917·6-s − 0.681·7-s + 0.833·8-s + 0.333·9-s − 2.87·10-s − 0.301·11-s − 0.880·12-s − 1.14·13-s − 1.08·14-s + 1.04·15-s − 0.200·16-s − 0.814·17-s + 0.529·18-s − 0.229·19-s − 2.76·20-s + 0.393·21-s − 0.479·22-s + 1.52·23-s − 0.481·24-s + 2.27·25-s − 1.82·26-s − 0.192·27-s − 1.03·28-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: −1-1
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: 11
Selberg data: (2, 627, ( :1/2), −1)(2,\ 627,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1−2.24T+2T2 1 - 2.24T + 2T^{2}
5 1+4.04T+5T2 1 + 4.04T + 5T^{2}
7 1+1.80T+7T2 1 + 1.80T + 7T^{2}
13 1+4.13T+13T2 1 + 4.13T + 13T^{2}
17 1+3.35T+17T2 1 + 3.35T + 17T^{2}
23 1−7.31T+23T2 1 - 7.31T + 23T^{2}
29 1−8.00T+29T2 1 - 8.00T + 29T^{2}
31 1+7.51T+31T2 1 + 7.51T + 31T^{2}
37 1+4.50T+37T2 1 + 4.50T + 37T^{2}
41 1−12.0T+41T2 1 - 12.0T + 41T^{2}
43 1+12.4T+43T2 1 + 12.4T + 43T^{2}
47 1+4.02T+47T2 1 + 4.02T + 47T^{2}
53 1+10.6T+53T2 1 + 10.6T + 53T^{2}
59 1−6.51T+59T2 1 - 6.51T + 59T^{2}
61 1+4.66T+61T2 1 + 4.66T + 61T^{2}
67 1−6.80T+67T2 1 - 6.80T + 67T^{2}
71 1−5.03T+71T2 1 - 5.03T + 71T^{2}
73 1−14.2T+73T2 1 - 14.2T + 73T^{2}
79 1+3.13T+79T2 1 + 3.13T + 79T^{2}
83 1−3.82T+83T2 1 - 3.82T + 83T^{2}
89 1+6.99T+89T2 1 + 6.99T + 89T^{2}
97 1+3.25T+97T2 1 + 3.25T + 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

−10.78651454765059016202610138905, −9.354676318847034742134288127395, −8.103577223997899842040001601549, −6.98756806927920459087440570584, −6.66335968344755607130141174577, −5.13631363302847026162632205350, −4.64494150598860412244688752899, −3.66273869984981939087818941652, −2.79510487889179487783268245210, 0, 2.79510487889179487783268245210, 3.66273869984981939087818941652, 4.64494150598860412244688752899, 5.13631363302847026162632205350, 6.66335968344755607130141174577, 6.98756806927920459087440570584, 8.103577223997899842040001601549, 9.354676318847034742134288127395, 10.78651454765059016202610138905

Graph of the ZZ-function along the critical line