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Properties

Label 2-357-1.1-c1-0-4
Degree 22
Conductor 357357
Sign 11
Analytic cond. 2.850652.85065
Root an. cond. 1.688381.68838
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  + 0.470·2-s + 3-s − 1.77·4-s + 0.529·5-s + 0.470·6-s + 7-s − 1.77·8-s + 9-s + 0.249·10-s + 4.77·11-s − 1.77·12-s + 5.02·13-s + 0.470·14-s + 0.529·15-s + 2.71·16-s + 17-s + 0.470·18-s − 1.47·19-s − 0.941·20-s + 21-s + 2.24·22-s − 0.778·23-s − 1.77·24-s − 4.71·25-s + 2.36·26-s + 27-s − 1.77·28-s + ⋯
L(s)  = 1  + 0.332·2-s + 0.577·3-s − 0.889·4-s + 0.236·5-s + 0.192·6-s + 0.377·7-s − 0.628·8-s + 0.333·9-s + 0.0787·10-s + 1.44·11-s − 0.513·12-s + 1.39·13-s + 0.125·14-s + 0.136·15-s + 0.679·16-s + 0.242·17-s + 0.110·18-s − 0.337·19-s − 0.210·20-s + 0.218·21-s + 0.479·22-s − 0.162·23-s − 0.363·24-s − 0.943·25-s + 0.464·26-s + 0.192·27-s − 0.336·28-s + ⋯

Functional equation

Λ(s)=(357s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(357s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 357357    =    3⋅7⋅173 \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 2.850652.85065
Root analytic conductor: 1.688381.68838
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 357, ( :1/2), 1)(2,\ 357,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7882943371.788294337
L(12)L(\frac12) ≈\approx 1.7882943371.788294337
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
7 1−T 1 - T
17 1−T 1 - T
good2 1−0.470T+2T2 1 - 0.470T + 2T^{2}
5 1−0.529T+5T2 1 - 0.529T + 5T^{2}
11 1−4.77T+11T2 1 - 4.77T + 11T^{2}
13 1−5.02T+13T2 1 - 5.02T + 13T^{2}
19 1+1.47T+19T2 1 + 1.47T + 19T^{2}
23 1+0.778T+23T2 1 + 0.778T + 23T^{2}
29 1+3.55T+29T2 1 + 3.55T + 29T^{2}
31 1+1.75T+31T2 1 + 1.75T + 31T^{2}
37 1+9.80T+37T2 1 + 9.80T + 37T^{2}
41 1−4.52T+41T2 1 - 4.52T + 41T^{2}
43 1+2.66T+43T2 1 + 2.66T + 43T^{2}
47 1+4.13T+47T2 1 + 4.13T + 47T^{2}
53 1−13.2T+53T2 1 - 13.2T + 53T^{2}
59 1+8.86T+59T2 1 + 8.86T + 59T^{2}
61 1+12.7T+61T2 1 + 12.7T + 61T^{2}
67 1−4.82T+67T2 1 - 4.82T + 67T^{2}
71 1−11.9T+71T2 1 - 11.9T + 71T^{2}
73 1+12.6T+73T2 1 + 12.6T + 73T^{2}
79 1−3.30T+79T2 1 - 3.30T + 79T^{2}
83 1−2.44T+83T2 1 - 2.44T + 83T^{2}
89 1−8.05T+89T2 1 - 8.05T + 89T^{2}
97 1−3.55T+97T2 1 - 3.55T + 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

−11.59485497294390259704713401527, −10.44978117422502264167856176870, −9.311964032565132756369545544178, −8.845872608504172513453441260467, −7.925336326812937879737117327477, −6.51658074289486414126472186181, −5.54158756840717382234083783981, −4.16104958825721235092395451518, −3.54178766549477272067615419805, −1.54369465786752064082182784273, 1.54369465786752064082182784273, 3.54178766549477272067615419805, 4.16104958825721235092395451518, 5.54158756840717382234083783981, 6.51658074289486414126472186181, 7.925336326812937879737117327477, 8.845872608504172513453441260467, 9.311964032565132756369545544178, 10.44978117422502264167856176870, 11.59485497294390259704713401527

Graph of the ZZ-function along the critical line