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Properties

Label 2-10e3-40.19-c0-0-0
Degree 22
Conductor 10001000
Sign 11
Analytic cond. 0.4990650.499065
Root an. cond. 0.7064450.706445
Motivic weight 00
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 − 0.618·7-s − 8-s + 9-s − 1.61·11-s + 1.61·13-s + 0.618·14-s + 16-s − 18-s + 0.618·19-s + 1.61·22-s + 1.61·23-s − 1.61·26-s − 0.618·28-s − 32-s + 36-s − 0.618·37-s − 0.618·38-s + 0.618·41-s − 1.61·44-s − 1.61·46-s + 1.61·47-s − 0.618·49-s + 1.61·52-s − 0.618·53-s + 0.618·56-s + ⋯
L(s)  = 1  − 2-s + 4-s − 0.618·7-s − 8-s + 9-s − 1.61·11-s + 1.61·13-s + 0.618·14-s + 16-s − 18-s + 0.618·19-s + 1.61·22-s + 1.61·23-s − 1.61·26-s − 0.618·28-s − 32-s + 36-s − 0.618·37-s − 0.618·38-s + 0.618·41-s − 1.61·44-s − 1.61·46-s + 1.61·47-s − 0.618·49-s + 1.61·52-s − 0.618·53-s + 0.618·56-s + ⋯

Functional equation

Λ(s)=(1000s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(1000s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10001000    =    23⋅532^{3} \cdot 5^{3}
Sign: 11
Analytic conductor: 0.4990650.499065
Root analytic conductor: 0.7064450.706445
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1000(499,⋅)\chi_{1000} (499, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1000, ( :0), 1)(2,\ 1000,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.67679919910.6767991991
L(12)L(\frac12) ≈\approx 0.67679919910.6767991991
L(1)L(1) 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
5 1 1
good3 1−T2 1 - T^{2}
7 1+0.618T+T2 1 + 0.618T + T^{2}
11 1+1.61T+T2 1 + 1.61T + T^{2}
13 1−1.61T+T2 1 - 1.61T + T^{2}
17 1−T2 1 - T^{2}
19 1−0.618T+T2 1 - 0.618T + T^{2}
23 1−1.61T+T2 1 - 1.61T + T^{2}
29 1−T2 1 - T^{2}
31 1−T2 1 - T^{2}
37 1+0.618T+T2 1 + 0.618T + T^{2}
41 1−0.618T+T2 1 - 0.618T + T^{2}
43 1−T2 1 - T^{2}
47 1−1.61T+T2 1 - 1.61T + T^{2}
53 1+0.618T+T2 1 + 0.618T + T^{2}
59 1−0.618T+T2 1 - 0.618T + T^{2}
61 1−T2 1 - T^{2}
67 1−T2 1 - T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+1.61T+T2 1 + 1.61T + T^{2}
97 1−T2 1 - T^{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

−10.21909375937087143863763296476, −9.336281646077878582184375078988, −8.606313541949706043374262076575, −7.68557501242738725718451301627, −7.03934487728620344713907360437, −6.12132479717912161917720685954, −5.15686831524762743437841247795, −3.62933963483017492191714838091, −2.67523889781754086218996491939, −1.18834119397106713934378917459, 1.18834119397106713934378917459, 2.67523889781754086218996491939, 3.62933963483017492191714838091, 5.15686831524762743437841247795, 6.12132479717912161917720685954, 7.03934487728620344713907360437, 7.68557501242738725718451301627, 8.606313541949706043374262076575, 9.336281646077878582184375078988, 10.21909375937087143863763296476

Graph of the ZZ-function along the critical line