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Properties

Label 2-20e2-25.19-c1-0-5
Degree 22
Conductor 400400
Sign 0.320−0.947i0.320 - 0.947i
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  + (1.31 + 1.81i)3-s + (2.19 + 0.432i)5-s + 1.85i·7-s + (−0.624 + 1.92i)9-s + (0.681 + 2.09i)11-s + (−4.55 − 1.47i)13-s + (2.10 + 4.54i)15-s + (3.76 − 5.17i)17-s + (−3.01 − 2.18i)19-s + (−3.37 + 2.44i)21-s + (0.905 − 0.294i)23-s + (4.62 + 1.89i)25-s + (2.08 − 0.677i)27-s + (−4.71 + 3.42i)29-s + (−5.42 − 3.94i)31-s + ⋯
L(s)  = 1  + (0.760 + 1.04i)3-s + (0.981 + 0.193i)5-s + 0.702i·7-s + (−0.208 + 0.640i)9-s + (0.205 + 0.632i)11-s + (−1.26 − 0.410i)13-s + (0.543 + 1.17i)15-s + (0.912 − 1.25i)17-s + (−0.690 − 0.501i)19-s + (−0.735 + 0.534i)21-s + (0.188 − 0.0613i)23-s + (0.925 + 0.379i)25-s + (0.401 − 0.130i)27-s + (−0.875 + 0.635i)29-s + (−0.974 − 0.708i)31-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((0.320−0.947i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+1/2) L(s)=((0.320−0.947i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 0.320−0.947i0.320 - 0.947i
Analytic conductor: 3.194013.19401
Root analytic conductor: 1.787181.78718
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ400(369,⋅)\chi_{400} (369, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :1/2), 0.320−0.947i)(2,\ 400,\ (\ :1/2),\ 0.320 - 0.947i)

Particular Values

L(1)L(1) ≈\approx 1.58770+1.13864i1.58770 + 1.13864i
L(12)L(\frac12) ≈\approx 1.58770+1.13864i1.58770 + 1.13864i
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+(−2.19−0.432i)T 1 + (-2.19 - 0.432i)T
good3 1+(−1.31−1.81i)T+(−0.927+2.85i)T2 1 + (-1.31 - 1.81i)T + (-0.927 + 2.85i)T^{2}
7 1−1.85iT−7T2 1 - 1.85iT - 7T^{2}
11 1+(−0.681−2.09i)T+(−8.89+6.46i)T2 1 + (-0.681 - 2.09i)T + (-8.89 + 6.46i)T^{2}
13 1+(4.55+1.47i)T+(10.5+7.64i)T2 1 + (4.55 + 1.47i)T + (10.5 + 7.64i)T^{2}
17 1+(−3.76+5.17i)T+(−5.25−16.1i)T2 1 + (-3.76 + 5.17i)T + (-5.25 - 16.1i)T^{2}
19 1+(3.01+2.18i)T+(5.87+18.0i)T2 1 + (3.01 + 2.18i)T + (5.87 + 18.0i)T^{2}
23 1+(−0.905+0.294i)T+(18.6−13.5i)T2 1 + (-0.905 + 0.294i)T + (18.6 - 13.5i)T^{2}
29 1+(4.71−3.42i)T+(8.96−27.5i)T2 1 + (4.71 - 3.42i)T + (8.96 - 27.5i)T^{2}
31 1+(5.42+3.94i)T+(9.57+29.4i)T2 1 + (5.42 + 3.94i)T + (9.57 + 29.4i)T^{2}
37 1+(−9.69−3.14i)T+(29.9+21.7i)T2 1 + (-9.69 - 3.14i)T + (29.9 + 21.7i)T^{2}
41 1+(1.82−5.63i)T+(−33.1−24.0i)T2 1 + (1.82 - 5.63i)T + (-33.1 - 24.0i)T^{2}
43 1−3.96iT−43T2 1 - 3.96iT - 43T^{2}
47 1+(5.93+8.16i)T+(−14.5+44.6i)T2 1 + (5.93 + 8.16i)T + (-14.5 + 44.6i)T^{2}
53 1+(2.46+3.39i)T+(−16.3+50.4i)T2 1 + (2.46 + 3.39i)T + (-16.3 + 50.4i)T^{2}
59 1+(1.86−5.73i)T+(−47.7−34.6i)T2 1 + (1.86 - 5.73i)T + (-47.7 - 34.6i)T^{2}
61 1+(1.13+3.49i)T+(−49.3+35.8i)T2 1 + (1.13 + 3.49i)T + (-49.3 + 35.8i)T^{2}
67 1+(−8.45+11.6i)T+(−20.7−63.7i)T2 1 + (-8.45 + 11.6i)T + (-20.7 - 63.7i)T^{2}
71 1+(−7.82+5.68i)T+(21.9−67.5i)T2 1 + (-7.82 + 5.68i)T + (21.9 - 67.5i)T^{2}
73 1+(−3.43+1.11i)T+(59.0−42.9i)T2 1 + (-3.43 + 1.11i)T + (59.0 - 42.9i)T^{2}
79 1+(9.13−6.63i)T+(24.4−75.1i)T2 1 + (9.13 - 6.63i)T + (24.4 - 75.1i)T^{2}
83 1+(−4.08+5.62i)T+(−25.6−78.9i)T2 1 + (-4.08 + 5.62i)T + (-25.6 - 78.9i)T^{2}
89 1+(3.23+9.96i)T+(−72.0+52.3i)T2 1 + (3.23 + 9.96i)T + (-72.0 + 52.3i)T^{2}
97 1+(−0.275−0.379i)T+(−29.9+92.2i)T2 1 + (-0.275 - 0.379i)T + (-29.9 + 92.2i)T^{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

−11.30168013240661315648785233189, −10.15078168810430520494440893864, −9.513329720530762687329862304217, −9.218811735828105961461063354142, −7.88448105584927543759585836561, −6.75234320474159363613566342902, −5.40022712144954876108510453652, −4.69287424095010337300717996346, −3.14067806977795165504308495475, −2.28514943398822762801082156367, 1.41197748728801755065443900511, 2.45193901396339050855241719125, 3.92728542943427841604379899167, 5.48158322332265141850482173242, 6.49476245249323919896548169197, 7.42957172380289241122512258791, 8.208806364608772261606919833156, 9.204636222119928996880952907929, 10.08515944721085353085846536465, 10.99465325101487791731271604438

Graph of the ZZ-function along the critical line