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Properties

Label 2-4000-40.29-c1-0-10
Degree 22
Conductor 40004000
Sign −0.937−0.349i-0.937 - 0.349i
Analytic cond. 31.940131.9401
Root an. cond. 5.651565.65156
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.83·3-s + 1.31i·7-s + 0.368·9-s + 6.60i·11-s − 2.96·13-s − 2.69i·17-s + 4.97i·19-s + 2.41i·21-s − 3.73i·23-s − 4.83·27-s − 5.52i·29-s − 8.36·31-s + 12.1i·33-s − 7.19·37-s − 5.44·39-s + ⋯
L(s)  = 1  + 1.05·3-s + 0.497i·7-s + 0.122·9-s + 1.99i·11-s − 0.822·13-s − 0.653i·17-s + 1.14i·19-s + 0.526i·21-s − 0.779i·23-s − 0.929·27-s − 1.02i·29-s − 1.50·31-s + 2.10i·33-s − 1.18·37-s − 0.871·39-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=((−0.937−0.349i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s+1/2) L(s)=((−0.937−0.349i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: −0.937−0.349i-0.937 - 0.349i
Analytic conductor: 31.940131.9401
Root analytic conductor: 5.651565.65156
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4000(3249,⋅)\chi_{4000} (3249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :1/2), −0.937−0.349i)(2,\ 4000,\ (\ :1/2),\ -0.937 - 0.349i)

Particular Values

L(1)L(1) ≈\approx 1.0595556891.059555689
L(12)L(\frac12) ≈\approx 1.0595556891.059555689
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 1−1.83T+3T2 1 - 1.83T + 3T^{2}
7 1−1.31iT−7T2 1 - 1.31iT - 7T^{2}
11 1−6.60iT−11T2 1 - 6.60iT - 11T^{2}
13 1+2.96T+13T2 1 + 2.96T + 13T^{2}
17 1+2.69iT−17T2 1 + 2.69iT - 17T^{2}
19 1−4.97iT−19T2 1 - 4.97iT - 19T^{2}
23 1+3.73iT−23T2 1 + 3.73iT - 23T^{2}
29 1+5.52iT−29T2 1 + 5.52iT - 29T^{2}
31 1+8.36T+31T2 1 + 8.36T + 31T^{2}
37 1+7.19T+37T2 1 + 7.19T + 37T^{2}
41 1+3.77T+41T2 1 + 3.77T + 41T^{2}
43 1−3.07T+43T2 1 - 3.07T + 43T^{2}
47 1−8.77iT−47T2 1 - 8.77iT - 47T^{2}
53 1−0.0464T+53T2 1 - 0.0464T + 53T^{2}
59 1+1.02iT−59T2 1 + 1.02iT - 59T^{2}
61 1+5.23iT−61T2 1 + 5.23iT - 61T^{2}
67 1+10.8T+67T2 1 + 10.8T + 67T^{2}
71 1+9.35T+71T2 1 + 9.35T + 71T^{2}
73 1+12.4iT−73T2 1 + 12.4iT - 73T^{2}
79 1−1.43T+79T2 1 - 1.43T + 79T^{2}
83 1−13.0T+83T2 1 - 13.0T + 83T^{2}
89 1+3.94T+89T2 1 + 3.94T + 89T^{2}
97 1−1.00iT−97T2 1 - 1.00iT - 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

−8.988020713727945783066177076174, −7.82104731018427409906300698164, −7.60984625798170381980020174938, −6.78145830510937831189843846255, −5.76123090026713373513833624323, −4.90502016626414341744351006627, −4.20037869614001222795844609501, −3.23053932305011817361526909697, −2.27949658256324284503006077260, −1.88332490497894619239394099518, 0.23075089552270524924905819615, 1.61168736959472412720006747829, 2.75440784689415094341436710063, 3.37530040623848461754143949642, 3.98104920087643103623861789145, 5.26007307465514856037528404791, 5.76265952973458916114202412483, 6.92566664334379138470708631949, 7.42084083590489407245530206057, 8.303924271145831696559357579450

Graph of the ZZ-function along the critical line