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Properties

Label 2-4000-40.29-c1-0-75
Degree 22
Conductor 40004000
Sign −0.156+0.987i-0.156 + 0.987i
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  + 0.939·3-s − 1.90i·7-s − 2.11·9-s − 2.70i·11-s + 5.75·13-s − 6.65i·17-s + 7.45i·19-s − 1.78i·21-s − 0.220i·23-s − 4.80·27-s − 4.62i·29-s + 5.76·31-s − 2.54i·33-s + 3.84·37-s + 5.40·39-s + ⋯
L(s)  = 1  + 0.542·3-s − 0.718i·7-s − 0.705·9-s − 0.816i·11-s + 1.59·13-s − 1.61i·17-s + 1.71i·19-s − 0.389i·21-s − 0.0460i·23-s − 0.925·27-s − 0.858i·29-s + 1.03·31-s − 0.442i·33-s + 0.631·37-s + 0.866·39-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=((−0.156+0.987i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s+1/2) L(s)=((−0.156+0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: −0.156+0.987i-0.156 + 0.987i
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), −0.156+0.987i)(2,\ 4000,\ (\ :1/2),\ -0.156 + 0.987i)

Particular Values

L(1)L(1) ≈\approx 1.8937809251.893780925
L(12)L(\frac12) ≈\approx 1.8937809251.893780925
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 1−0.939T+3T2 1 - 0.939T + 3T^{2}
7 1+1.90iT−7T2 1 + 1.90iT - 7T^{2}
11 1+2.70iT−11T2 1 + 2.70iT - 11T^{2}
13 1−5.75T+13T2 1 - 5.75T + 13T^{2}
17 1+6.65iT−17T2 1 + 6.65iT - 17T^{2}
19 1−7.45iT−19T2 1 - 7.45iT - 19T^{2}
23 1+0.220iT−23T2 1 + 0.220iT - 23T^{2}
29 1+4.62iT−29T2 1 + 4.62iT - 29T^{2}
31 1−5.76T+31T2 1 - 5.76T + 31T^{2}
37 1−3.84T+37T2 1 - 3.84T + 37T^{2}
41 1+7.94T+41T2 1 + 7.94T + 41T^{2}
43 1+10.3T+43T2 1 + 10.3T + 43T^{2}
47 1−1.41iT−47T2 1 - 1.41iT - 47T^{2}
53 1−3.09T+53T2 1 - 3.09T + 53T^{2}
59 1+5.84iT−59T2 1 + 5.84iT - 59T^{2}
61 1+11.6iT−61T2 1 + 11.6iT - 61T^{2}
67 1−5.42T+67T2 1 - 5.42T + 67T^{2}
71 1+13.8T+71T2 1 + 13.8T + 71T^{2}
73 1−0.290iT−73T2 1 - 0.290iT - 73T^{2}
79 1−0.354T+79T2 1 - 0.354T + 79T^{2}
83 1+6.09T+83T2 1 + 6.09T + 83T^{2}
89 1+6.94T+89T2 1 + 6.94T + 89T^{2}
97 1+4.56iT−97T2 1 + 4.56iT - 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

−8.289695222796190468873951757314, −7.77117606757400693635253744991, −6.72644042314634762557727161118, −6.05788709597503336598916312873, −5.37659353113220329055261298903, −4.24737357852047744413683973699, −3.46984831601876945892241829872, −2.95184813023533195963059492292, −1.63769700447174123732600010927, −0.51232105410223067258114410832, 1.34368856260675667863694771799, 2.33698606593799472661192664619, 3.14327026904573963848826041789, 3.96111520846372738742720166447, 4.90126262050381607639024534017, 5.78347866603507020274545538478, 6.39204176793346681640410525562, 7.17320189998129566408721519255, 8.251592291331737842387369124574, 8.670419966673517039775976522712

Graph of the ZZ-function along the critical line