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Properties

Label 2-40e2-20.7-c1-0-3
Degree 22
Conductor 16001600
Sign 0.525−0.850i0.525 - 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: 0.525−0.850i0.525 - 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 0.95500789100.9550078910
L(12)L(\frac12) ≈\approx 0.95500789100.9550078910
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

−9.435330880818362861898421988574, −8.977489032846631961492653431021, −7.54279911710181256328349125209, −6.99585954222057837778936537293, −6.74290309768572867376203147843, −5.15012851878892312761354036258, −4.88516503467094398560240983087, −3.62600451083269632399138749568, −2.26110268668030789283699228157, −1.21720050887824962521273557664, 0.42983496778959672616558086028, 2.22956301135360237128084143282, 3.23976030722810859147189041380, 4.46462991134290027235382930180, 5.19295934726610047977433885556, 5.79225364350964496423268846699, 6.73297676475925844216356226049, 7.970434541080482739222639776938, 8.310488556918893265567780529407, 9.403435773382798172488660615074

Graph of the ZZ-function along the critical line