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Properties

Label 2-4000-1.1-c1-0-19
Degree 22
Conductor 40004000
Sign 11
Analytic cond. 31.940131.9401
Root an. cond. 5.651565.65156
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  − 2.29·3-s + 0.938·7-s + 2.25·9-s + 1.16·11-s + 2.31·13-s − 6.64·17-s + 5.70·19-s − 2.15·21-s + 1.63·23-s + 1.70·27-s − 1.93·29-s + 4.77·31-s − 2.66·33-s − 8.06·37-s − 5.31·39-s + 4.70·41-s + 3.78·43-s − 1.00·47-s − 6.11·49-s + 15.2·51-s + 4.99·53-s − 13.0·57-s + 5.79·59-s − 2.55·61-s + 2.11·63-s − 11.9·67-s − 3.75·69-s + ⋯
L(s)  = 1  − 1.32·3-s + 0.354·7-s + 0.751·9-s + 0.350·11-s + 0.642·13-s − 1.61·17-s + 1.30·19-s − 0.469·21-s + 0.341·23-s + 0.328·27-s − 0.359·29-s + 0.857·31-s − 0.464·33-s − 1.32·37-s − 0.850·39-s + 0.734·41-s + 0.577·43-s − 0.146·47-s − 0.874·49-s + 2.13·51-s + 0.685·53-s − 1.73·57-s + 0.753·59-s − 0.327·61-s + 0.266·63-s − 1.45·67-s − 0.451·69-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 11
Analytic conductor: 31.940131.9401
Root analytic conductor: 5.651565.65156
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4000, ( :1/2), 1)(2,\ 4000,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1193854301.119385430
L(12)L(\frac12) ≈\approx 1.1193854301.119385430
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+2.29T+3T2 1 + 2.29T + 3T^{2}
7 1−0.938T+7T2 1 - 0.938T + 7T^{2}
11 1−1.16T+11T2 1 - 1.16T + 11T^{2}
13 1−2.31T+13T2 1 - 2.31T + 13T^{2}
17 1+6.64T+17T2 1 + 6.64T + 17T^{2}
19 1−5.70T+19T2 1 - 5.70T + 19T^{2}
23 1−1.63T+23T2 1 - 1.63T + 23T^{2}
29 1+1.93T+29T2 1 + 1.93T + 29T^{2}
31 1−4.77T+31T2 1 - 4.77T + 31T^{2}
37 1+8.06T+37T2 1 + 8.06T + 37T^{2}
41 1−4.70T+41T2 1 - 4.70T + 41T^{2}
43 1−3.78T+43T2 1 - 3.78T + 43T^{2}
47 1+1.00T+47T2 1 + 1.00T + 47T^{2}
53 1−4.99T+53T2 1 - 4.99T + 53T^{2}
59 1−5.79T+59T2 1 - 5.79T + 59T^{2}
61 1+2.55T+61T2 1 + 2.55T + 61T^{2}
67 1+11.9T+67T2 1 + 11.9T + 67T^{2}
71 1−3.85T+71T2 1 - 3.85T + 71T^{2}
73 1−0.210T+73T2 1 - 0.210T + 73T^{2}
79 1+14.6T+79T2 1 + 14.6T + 79T^{2}
83 1−5.62T+83T2 1 - 5.62T + 83T^{2}
89 1−10.8T+89T2 1 - 10.8T + 89T^{2}
97 1+14.5T+97T2 1 + 14.5T + 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

−8.570675723822496990597371436187, −7.52525679430210814515224531425, −6.81689232367689618844061990005, −6.21315727204093365589448252854, −5.49786852370569522001052843925, −4.81139319309716281495186953591, −4.10198971501455955248780041584, −2.98662624847037361550114663755, −1.70145751572325419520680964013, −0.67003489901495107121056442664, 0.67003489901495107121056442664, 1.70145751572325419520680964013, 2.98662624847037361550114663755, 4.10198971501455955248780041584, 4.81139319309716281495186953591, 5.49786852370569522001052843925, 6.21315727204093365589448252854, 6.81689232367689618844061990005, 7.52525679430210814515224531425, 8.570675723822496990597371436187

Graph of the ZZ-function along the critical line