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Properties

Label 2-4000-5.4-c1-0-21
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 no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.40i·3-s − 2.20i·7-s − 2.78·9-s − 1.68·11-s + 6.87i·13-s + 5.86i·17-s + 0.108·19-s − 5.31·21-s − 4.40i·23-s − 0.513i·27-s + 4.25·29-s − 9.07·31-s + 4.04i·33-s − 1.74i·37-s + 16.5·39-s + ⋯
L(s)  = 1  − 1.38i·3-s − 0.834i·7-s − 0.928·9-s − 0.507·11-s + 1.90i·13-s + 1.42i·17-s + 0.0248·19-s − 1.15·21-s − 0.918i·23-s − 0.0987i·27-s + 0.790·29-s − 1.62·31-s + 0.704i·33-s − 0.287i·37-s + 2.64·39-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 & \, \overline{\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 & \, \overline{\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: χ4000(1249,⋅)\chi_{4000} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :1/2), 1)(2,\ 4000,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4123255371.412325537
L(12)L(\frac12) ≈\approx 1.4123255371.412325537
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+2.40iT−3T2 1 + 2.40iT - 3T^{2}
7 1+2.20iT−7T2 1 + 2.20iT - 7T^{2}
11 1+1.68T+11T2 1 + 1.68T + 11T^{2}
13 1−6.87iT−13T2 1 - 6.87iT - 13T^{2}
17 1−5.86iT−17T2 1 - 5.86iT - 17T^{2}
19 1−0.108T+19T2 1 - 0.108T + 19T^{2}
23 1+4.40iT−23T2 1 + 4.40iT - 23T^{2}
29 1−4.25T+29T2 1 - 4.25T + 29T^{2}
31 1+9.07T+31T2 1 + 9.07T + 31T^{2}
37 1+1.74iT−37T2 1 + 1.74iT - 37T^{2}
41 1−7.95T+41T2 1 - 7.95T + 41T^{2}
43 1−8.89iT−43T2 1 - 8.89iT - 43T^{2}
47 1−10.5iT−47T2 1 - 10.5iT - 47T^{2}
53 1−8.51iT−53T2 1 - 8.51iT - 53T^{2}
59 1−7.67T+59T2 1 - 7.67T + 59T^{2}
61 1−1.94T+61T2 1 - 1.94T + 61T^{2}
67 1−0.788iT−67T2 1 - 0.788iT - 67T^{2}
71 1+5.48T+71T2 1 + 5.48T + 71T^{2}
73 1−11.9iT−73T2 1 - 11.9iT - 73T^{2}
79 1−1.28T+79T2 1 - 1.28T + 79T^{2}
83 1−4.08iT−83T2 1 - 4.08iT - 83T^{2}
89 1−2.37T+89T2 1 - 2.37T + 89T^{2}
97 1+5.70iT−97T2 1 + 5.70iT - 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

−8.275482766501717128960074220884, −7.56838609830062879538936439046, −7.03379101006718697980338715534, −6.41099484858286476777829854740, −5.82086821730217476235371254793, −4.45663093316803427550304183717, −4.02393573374010130512224018090, −2.64809713064670485827720827647, −1.81879063099329373397517166208, −1.03686488130394859177311327608, 0.45727767090307969466642449951, 2.30033042211614620867202342961, 3.14488390758048170810231885965, 3.70959877584956445409211120042, 4.94381600606945568338396978626, 5.32210404247796694966294989365, 5.78408324917380360710216193453, 7.13823717639861220534380707361, 7.78329770045127948457860160551, 8.711667353335464571468619606713

Graph of the ZZ-function along the critical line