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Properties

Label 2-40e2-20.7-c1-0-30
Degree 22
Conductor 16001600
Sign −0.525+0.850i-0.525 + 0.850i
Analytic cond. 12.776012.7760
Root an. cond. 3.574363.57436
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 + i)3-s + (−1 + i)7-s − i·9-s − 4i·11-s + (−4 + 4i)13-s + (−4 − 4i)17-s − 4·19-s − 2·21-s + (−5 − 5i)23-s + (4 − 4i)27-s + 2i·29-s − 8i·31-s + (4 − 4i)33-s − 8·39-s − 4·41-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (−0.377 + 0.377i)7-s − 0.333i·9-s − 1.20i·11-s + (−1.10 + 1.10i)13-s + (−0.970 − 0.970i)17-s − 0.917·19-s − 0.436·21-s + (−1.04 − 1.04i)23-s + (0.769 − 0.769i)27-s + 0.371i·29-s − 1.43i·31-s + (0.696 − 0.696i)33-s − 1.28·39-s − 0.624·41-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((−0.525+0.850i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=((−0.525+0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: −0.525+0.850i-0.525 + 0.850i
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1600(1407,⋅)\chi_{1600} (1407, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :1/2), −0.525+0.850i)(2,\ 1600,\ (\ :1/2),\ -0.525 + 0.850i)

Particular Values

L(1)L(1) ≈\approx 0.60583118480.6058311848
L(12)L(\frac12) ≈\approx 0.60583118480.6058311848
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−i)T+3iT2 1 + (-1 - i)T + 3iT^{2}
7 1+(1−i)T−7iT2 1 + (1 - i)T - 7iT^{2}
11 1+4iT−11T2 1 + 4iT - 11T^{2}
13 1+(4−4i)T−13iT2 1 + (4 - 4i)T - 13iT^{2}
17 1+(4+4i)T+17iT2 1 + (4 + 4i)T + 17iT^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+(5+5i)T+23iT2 1 + (5 + 5i)T + 23iT^{2}
29 1−2iT−29T2 1 - 2iT - 29T^{2}
31 1+8iT−31T2 1 + 8iT - 31T^{2}
37 1+37iT2 1 + 37iT^{2}
41 1+4T+41T2 1 + 4T + 41T^{2}
43 1+(−7−7i)T+43iT2 1 + (-7 - 7i)T + 43iT^{2}
47 1+(3−3i)T−47iT2 1 + (3 - 3i)T - 47iT^{2}
53 1+(4−4i)T−53iT2 1 + (4 - 4i)T - 53iT^{2}
59 1+4T+59T2 1 + 4T + 59T^{2}
61 1−8T+61T2 1 - 8T + 61T^{2}
67 1+(3−3i)T−67iT2 1 + (3 - 3i)T - 67iT^{2}
71 1+16iT−71T2 1 + 16iT - 71T^{2}
73 1+(−4+4i)T−73iT2 1 + (-4 + 4i)T - 73iT^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+(5+5i)T+83iT2 1 + (5 + 5i)T + 83iT^{2}
89 1+10iT−89T2 1 + 10iT - 89T^{2}
97 1+(−12−12i)T+97iT2 1 + (-12 - 12i)T + 97iT^{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

−9.192900721160815985217300973106, −8.608609749428778522541462133173, −7.67151786192096691761061187536, −6.51233916545006477869457783845, −6.11105529258979221719144960840, −4.67765169022726407765509898463, −4.15087887146774025401614048191, −2.97357296964952183081668291802, −2.27037120910283167330980891325, −0.19487628435779945869852605905, 1.78525176044017764225093375829, 2.46311500184935181434349186261, 3.69070812349028980362855878864, 4.65910509730366936616086594755, 5.57524113552240322005604675061, 6.80379654127223642158764400275, 7.23639904030759459189969826805, 8.089710328410930518911263394814, 8.651729898430027713675274265469, 9.878476042868914690461753608833

Graph of the ZZ-function along the critical line