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Properties

Label 2-10e3-1.1-c1-0-2
Degree 22
Conductor 10001000
Sign 11
Analytic cond. 7.985047.98504
Root an. cond. 2.825782.82578
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  − 0.543·3-s − 2.11·7-s − 2.70·9-s − 4.99·11-s + 6.32·13-s + 3.75·17-s + 1.90·19-s + 1.14·21-s + 9.35·23-s + 3.09·27-s + 7.22·29-s + 0.994·31-s + 2.71·33-s − 6.37·37-s − 3.43·39-s + 4.18·41-s + 1.45·43-s + 2.78·47-s − 2.52·49-s − 2.04·51-s + 1.52·53-s − 1.03·57-s + 1.47·59-s + 8.79·61-s + 5.72·63-s − 12.0·67-s − 5.08·69-s + ⋯
L(s)  = 1  − 0.313·3-s − 0.799·7-s − 0.901·9-s − 1.50·11-s + 1.75·13-s + 0.911·17-s + 0.437·19-s + 0.250·21-s + 1.94·23-s + 0.596·27-s + 1.34·29-s + 0.178·31-s + 0.472·33-s − 1.04·37-s − 0.550·39-s + 0.653·41-s + 0.222·43-s + 0.406·47-s − 0.360·49-s − 0.285·51-s + 0.209·53-s − 0.137·57-s + 0.192·59-s + 1.12·61-s + 0.720·63-s − 1.47·67-s − 0.611·69-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 10001000    =    23⋅532^{3} \cdot 5^{3}
Sign: 11
Analytic conductor: 7.985047.98504
Root analytic conductor: 2.825782.82578
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1000, ( :1/2), 1)(2,\ 1000,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1930926621.193092662
L(12)L(\frac12) ≈\approx 1.1930926621.193092662
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 1
5 1 1
good3 1+0.543T+3T2 1 + 0.543T + 3T^{2}
7 1+2.11T+7T2 1 + 2.11T + 7T^{2}
11 1+4.99T+11T2 1 + 4.99T + 11T^{2}
13 1−6.32T+13T2 1 - 6.32T + 13T^{2}
17 1−3.75T+17T2 1 - 3.75T + 17T^{2}
19 1−1.90T+19T2 1 - 1.90T + 19T^{2}
23 1−9.35T+23T2 1 - 9.35T + 23T^{2}
29 1−7.22T+29T2 1 - 7.22T + 29T^{2}
31 1−0.994T+31T2 1 - 0.994T + 31T^{2}
37 1+6.37T+37T2 1 + 6.37T + 37T^{2}
41 1−4.18T+41T2 1 - 4.18T + 41T^{2}
43 1−1.45T+43T2 1 - 1.45T + 43T^{2}
47 1−2.78T+47T2 1 - 2.78T + 47T^{2}
53 1−1.52T+53T2 1 - 1.52T + 53T^{2}
59 1−1.47T+59T2 1 - 1.47T + 59T^{2}
61 1−8.79T+61T2 1 - 8.79T + 61T^{2}
67 1+12.0T+67T2 1 + 12.0T + 67T^{2}
71 1+12.3T+71T2 1 + 12.3T + 71T^{2}
73 1−9.09T+73T2 1 - 9.09T + 73T^{2}
79 1+8.91T+79T2 1 + 8.91T + 79T^{2}
83 1−12.1T+83T2 1 - 12.1T + 83T^{2}
89 1−3.08T+89T2 1 - 3.08T + 89T^{2}
97 1−10.1T+97T2 1 - 10.1T + 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

−10.19388816888478715002195662425, −9.001903722517744142311909790216, −8.438624430237509876577644486088, −7.46551736755786102655952818709, −6.42504344208708348142551815768, −5.69358268310058173865848038870, −4.95829748245161908284484670251, −3.38963219773857748805981466592, −2.83927531314506005597144705385, −0.867530540567614798367836764457, 0.867530540567614798367836764457, 2.83927531314506005597144705385, 3.38963219773857748805981466592, 4.95829748245161908284484670251, 5.69358268310058173865848038870, 6.42504344208708348142551815768, 7.46551736755786102655952818709, 8.438624430237509876577644486088, 9.001903722517744142311909790216, 10.19388816888478715002195662425

Graph of the ZZ-function along the critical line