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Properties

Label 2-5850-1.1-c1-0-44
Degree 22
Conductor 58505850
Sign 11
Analytic cond. 46.712446.7124
Root an. cond. 6.834656.83465
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  + 2-s + 4-s + 4.38·7-s + 8-s − 2·11-s − 13-s + 4.38·14-s + 16-s + 5.86·17-s − 0.973·19-s − 2·22-s + 7.79·23-s − 26-s + 4.38·28-s − 0.973·29-s − 1.79·31-s + 32-s + 5.86·34-s + 0.591·37-s − 0.973·38-s − 4.81·41-s − 4.68·43-s − 2·44-s + 7.79·46-s + 0.381·47-s + 12.1·49-s − 52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.65·7-s + 0.353·8-s − 0.603·11-s − 0.277·13-s + 1.17·14-s + 0.250·16-s + 1.42·17-s − 0.223·19-s − 0.426·22-s + 1.62·23-s − 0.196·26-s + 0.828·28-s − 0.180·29-s − 0.321·31-s + 0.176·32-s + 1.00·34-s + 0.0972·37-s − 0.157·38-s − 0.752·41-s − 0.714·43-s − 0.301·44-s + 1.14·46-s + 0.0556·47-s + 1.74·49-s − 0.138·52-s + ⋯

Functional equation

Λ(s)=(5850s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5850s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 58505850    =    2⋅32⋅52⋅132 \cdot 3^{2} \cdot 5^{2} \cdot 13
Sign: 11
Analytic conductor: 46.712446.7124
Root analytic conductor: 6.834656.83465
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5850, ( :1/2), 1)(2,\ 5850,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.1754954004.175495400
L(12)L(\frac12) ≈\approx 4.1754954004.175495400
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)
bad2 1−T 1 - T
3 1 1
5 1 1
13 1+T 1 + T
good7 1−4.38T+7T2 1 - 4.38T + 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
17 1−5.86T+17T2 1 - 5.86T + 17T^{2}
19 1+0.973T+19T2 1 + 0.973T + 19T^{2}
23 1−7.79T+23T2 1 - 7.79T + 23T^{2}
29 1+0.973T+29T2 1 + 0.973T + 29T^{2}
31 1+1.79T+31T2 1 + 1.79T + 31T^{2}
37 1−0.591T+37T2 1 - 0.591T + 37T^{2}
41 1+4.81T+41T2 1 + 4.81T + 41T^{2}
43 1+4.68T+43T2 1 + 4.68T + 43T^{2}
47 1−0.381T+47T2 1 - 0.381T + 47T^{2}
53 1−7.79T+53T2 1 - 7.79T + 53T^{2}
59 1−0.973T+59T2 1 - 0.973T + 59T^{2}
61 1+0.817T+61T2 1 + 0.817T + 61T^{2}
67 1+1.79T+67T2 1 + 1.79T + 67T^{2}
71 1−3.92T+71T2 1 - 3.92T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−10.9T+79T2 1 - 10.9T + 79T^{2}
83 1+6.97T+83T2 1 + 6.97T + 83T^{2}
89 1+0.973T+89T2 1 + 0.973T + 89T^{2}
97 1−18.6T+97T2 1 - 18.6T + 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

−7.981267485804404022571294126993, −7.42566328450048093725462329928, −6.74541771013221189530218980370, −5.57819601364454800205672567273, −5.21039040137060604947210738225, −4.66900664890338250431758766849, −3.71122162849884799228237559385, −2.85285923543981849146668516826, −1.93158311479120798124281007810, −1.04440546448079732842979602719, 1.04440546448079732842979602719, 1.93158311479120798124281007810, 2.85285923543981849146668516826, 3.71122162849884799228237559385, 4.66900664890338250431758766849, 5.21039040137060604947210738225, 5.57819601364454800205672567273, 6.74541771013221189530218980370, 7.42566328450048093725462329928, 7.981267485804404022571294126993

Graph of the ZZ-function along the critical line