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Properties

Label 2-40e2-20.7-c1-0-12
Degree 22
Conductor 16001600
Sign 0.850+0.525i0.850 + 0.525i
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 + (−3 + 3i)7-s − i·9-s + 6·21-s + (1 + i)23-s + (−4 + 4i)27-s − 6i·29-s + 12·41-s + (9 + 9i)43-s + (7 − 7i)47-s − 11i·49-s + 8·61-s + (3 + 3i)63-s + (3 − 3i)67-s − 2i·69-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (−1.13 + 1.13i)7-s − 0.333i·9-s + 1.30·21-s + (0.208 + 0.208i)23-s + (−0.769 + 0.769i)27-s − 1.11i·29-s + 1.87·41-s + (1.37 + 1.37i)43-s + (1.02 − 1.02i)47-s − 1.57i·49-s + 1.02·61-s + (0.377 + 0.377i)63-s + (0.366 − 0.366i)67-s − 0.240i·69-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 0.850+0.525i0.850 + 0.525i
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), 0.850+0.525i)(2,\ 1600,\ (\ :1/2),\ 0.850 + 0.525i)

Particular Values

L(1)L(1) ≈\approx 1.0391914061.039191406
L(12)L(\frac12) ≈\approx 1.0391914061.039191406
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+(3−3i)T−7iT2 1 + (3 - 3i)T - 7iT^{2}
11 1−11T2 1 - 11T^{2}
13 1−13iT2 1 - 13iT^{2}
17 1+17iT2 1 + 17iT^{2}
19 1+19T2 1 + 19T^{2}
23 1+(−1−i)T+23iT2 1 + (-1 - i)T + 23iT^{2}
29 1+6iT−29T2 1 + 6iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+37iT2 1 + 37iT^{2}
41 1−12T+41T2 1 - 12T + 41T^{2}
43 1+(−9−9i)T+43iT2 1 + (-9 - 9i)T + 43iT^{2}
47 1+(−7+7i)T−47iT2 1 + (-7 + 7i)T - 47iT^{2}
53 1−53iT2 1 - 53iT^{2}
59 1+59T2 1 + 59T^{2}
61 1−8T+61T2 1 - 8T + 61T^{2}
67 1+(−3+3i)T−67iT2 1 + (-3 + 3i)T - 67iT^{2}
71 1−71T2 1 - 71T^{2}
73 1−73iT2 1 - 73iT^{2}
79 1+79T2 1 + 79T^{2}
83 1+(11+11i)T+83iT2 1 + (11 + 11i)T + 83iT^{2}
89 1−6iT−89T2 1 - 6iT - 89T^{2}
97 1+97iT2 1 + 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

−9.378468980958517742437818312667, −8.676683686322070143775019249456, −7.58391356396562637697442744786, −6.79580121814763522725822710827, −5.92255289785252634124047291822, −5.73648311835224548715439573185, −4.29392024536477669311303045219, −3.17512173195372653193629663407, −2.24603293672649670424589853110, −0.66370069951966065907765649761, 0.78258933533769630609459500687, 2.56100977990067252553594394880, 3.74131552264969916001740768214, 4.35635188458020468337971909209, 5.38943781518919203046266981553, 6.18119996349557034845715173650, 7.09655167360036780894579923693, 7.67013848081013162138557875546, 8.908489870436310052046291335253, 9.604994132106875945435616786468

Graph of the ZZ-function along the critical line