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Properties

Label 2-40e2-16.5-c1-0-23
Degree 22
Conductor 16001600
Sign 0.453+0.891i0.453 + 0.891i
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  + (−1.99 + 1.99i)3-s − 1.09i·7-s − 4.93i·9-s + (2.33 + 2.33i)11-s + (−1.80 + 1.80i)13-s − 4.93·17-s + (2.03 − 2.03i)19-s + (2.17 + 2.17i)21-s − 1.45i·23-s + (3.84 + 3.84i)27-s + (0.707 − 0.707i)29-s − 10.1·31-s − 9.28·33-s + (−4.35 − 4.35i)37-s − 7.20i·39-s + ⋯
L(s)  = 1  + (−1.14 + 1.14i)3-s − 0.412i·7-s − 1.64i·9-s + (0.703 + 0.703i)11-s + (−0.501 + 0.501i)13-s − 1.19·17-s + (0.467 − 0.467i)19-s + (0.473 + 0.473i)21-s − 0.303i·23-s + (0.740 + 0.740i)27-s + (0.131 − 0.131i)29-s − 1.81·31-s − 1.61·33-s + (−0.715 − 0.715i)37-s − 1.15i·39-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((0.453+0.891i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=((0.453+0.891i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 0.453+0.891i0.453 + 0.891i
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1600(1201,⋅)\chi_{1600} (1201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :1/2), 0.453+0.891i)(2,\ 1600,\ (\ :1/2),\ 0.453 + 0.891i)

Particular Values

L(1)L(1) ≈\approx 0.46694886790.4669488679
L(12)L(\frac12) ≈\approx 0.46694886790.4669488679
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+(1.99−1.99i)T−3iT2 1 + (1.99 - 1.99i)T - 3iT^{2}
7 1+1.09iT−7T2 1 + 1.09iT - 7T^{2}
11 1+(−2.33−2.33i)T+11iT2 1 + (-2.33 - 2.33i)T + 11iT^{2}
13 1+(1.80−1.80i)T−13iT2 1 + (1.80 - 1.80i)T - 13iT^{2}
17 1+4.93T+17T2 1 + 4.93T + 17T^{2}
19 1+(−2.03+2.03i)T−19iT2 1 + (-2.03 + 2.03i)T - 19iT^{2}
23 1+1.45iT−23T2 1 + 1.45iT - 23T^{2}
29 1+(−0.707+0.707i)T−29iT2 1 + (-0.707 + 0.707i)T - 29iT^{2}
31 1+10.1T+31T2 1 + 10.1T + 31T^{2}
37 1+(4.35+4.35i)T+37iT2 1 + (4.35 + 4.35i)T + 37iT^{2}
41 1+10.2iT−41T2 1 + 10.2iT - 41T^{2}
43 1+(−2.22−2.22i)T+43iT2 1 + (-2.22 - 2.22i)T + 43iT^{2}
47 1+2.09T+47T2 1 + 2.09T + 47T^{2}
53 1+(−0.215−0.215i)T+53iT2 1 + (-0.215 - 0.215i)T + 53iT^{2}
59 1+(1.16+1.16i)T+59iT2 1 + (1.16 + 1.16i)T + 59iT^{2}
61 1+(3.46−3.46i)T−61iT2 1 + (3.46 - 3.46i)T - 61iT^{2}
67 1+(−5.04+5.04i)T−67iT2 1 + (-5.04 + 5.04i)T - 67iT^{2}
71 1+6.40iT−71T2 1 + 6.40iT - 71T^{2}
73 1−5.24iT−73T2 1 - 5.24iT - 73T^{2}
79 1−2.61T+79T2 1 - 2.61T + 79T^{2}
83 1+(−5.67+5.67i)T−83iT2 1 + (-5.67 + 5.67i)T - 83iT^{2}
89 1+6.87iT−89T2 1 + 6.87iT - 89T^{2}
97 1−3.77T+97T2 1 - 3.77T + 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

−9.285471527543659532716746492599, −8.961958665466370610776490098750, −7.34167734404972936684276479672, −6.83845443718898371491951429352, −5.87378207548115033497716419573, −4.99416473802035998163159548956, −4.35024892634040299698053282024, −3.68187021489925213461058129501, −2.02403006827326467510003018192, −0.23514989886664444665732613059, 1.12730592642447118092165094479, 2.20284899219634700520171004193, 3.53690632451609662125736864073, 4.90284548144179871980125288864, 5.64010161364021822435073745251, 6.31326331439637901503343100563, 7.00372134994133021365585656892, 7.75985640700335261478506007519, 8.665656914424141832089762723680, 9.490305423153045784012081280378

Graph of the ZZ-function along the critical line