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Properties

Label 2-20e2-20.7-c1-0-3
Degree 22
Conductor 400400
Sign 0.850+0.525i0.850 + 0.525i
Analytic cond. 3.194013.19401
Root an. cond. 1.787181.78718
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  − 3i·9-s + (5 − 5i)13-s + (5 + 5i)17-s − 4i·29-s + (−5 − 5i)37-s + 8·41-s + 7i·49-s + (−5 + 5i)53-s − 12·61-s + (−5 + 5i)73-s − 9·81-s + 16i·89-s + (−5 − 5i)97-s + 2·101-s − 6i·109-s + ⋯
L(s)  = 1  − i·9-s + (1.38 − 1.38i)13-s + (1.21 + 1.21i)17-s − 0.742i·29-s + (−0.821 − 0.821i)37-s + 1.24·41-s + i·49-s + (−0.686 + 0.686i)53-s − 1.53·61-s + (−0.585 + 0.585i)73-s − 81-s + 1.69i·89-s + (−0.507 − 0.507i)97-s + 0.199·101-s − 0.574i·109-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((0.850+0.525i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+1/2) L(s)=((0.850+0.525i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{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: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 0.850+0.525i0.850 + 0.525i
Analytic conductor: 3.194013.19401
Root analytic conductor: 1.787181.78718
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ400(207,⋅)\chi_{400} (207, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :1/2), 0.850+0.525i)(2,\ 400,\ (\ :1/2),\ 0.850 + 0.525i)

Particular Values

L(1)L(1) ≈\approx 1.37935−0.391845i1.37935 - 0.391845i
L(12)L(\frac12) ≈\approx 1.37935−0.391845i1.37935 - 0.391845i
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+3iT2 1 + 3iT^{2}
7 1−7iT2 1 - 7iT^{2}
11 1−11T2 1 - 11T^{2}
13 1+(−5+5i)T−13iT2 1 + (-5 + 5i)T - 13iT^{2}
17 1+(−5−5i)T+17iT2 1 + (-5 - 5i)T + 17iT^{2}
19 1+19T2 1 + 19T^{2}
23 1+23iT2 1 + 23iT^{2}
29 1+4iT−29T2 1 + 4iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+(5+5i)T+37iT2 1 + (5 + 5i)T + 37iT^{2}
41 1−8T+41T2 1 - 8T + 41T^{2}
43 1+43iT2 1 + 43iT^{2}
47 1−47iT2 1 - 47iT^{2}
53 1+(5−5i)T−53iT2 1 + (5 - 5i)T - 53iT^{2}
59 1+59T2 1 + 59T^{2}
61 1+12T+61T2 1 + 12T + 61T^{2}
67 1−67iT2 1 - 67iT^{2}
71 1−71T2 1 - 71T^{2}
73 1+(5−5i)T−73iT2 1 + (5 - 5i)T - 73iT^{2}
79 1+79T2 1 + 79T^{2}
83 1+83iT2 1 + 83iT^{2}
89 1−16iT−89T2 1 - 16iT - 89T^{2}
97 1+(5+5i)T+97iT2 1 + (5 + 5i)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

−11.04472555806920005307368319708, −10.38665363389198613782678020822, −9.368931877555665881972871881716, −8.394194096742075503759627841014, −7.61950321962881613564710904487, −6.17528400162867302190580578900, −5.71084731742201172430389379938, −4.00464382900555206770790773095, −3.15925999426796568586280927185, −1.13266468726042336887303247194, 1.61431408663841428470958862339, 3.19380337442781207459228011090, 4.50293949880647670934962873573, 5.52846652566638912601909530456, 6.68928335851737744391590006756, 7.64187653849720920038089449727, 8.629770810801943550226109079514, 9.493423312652212097735151495875, 10.54545353312228005749820967794, 11.34389581752958628359760485054

Graph of the ZZ-function along the critical line