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Properties

Label 2-20e2-100.27-c1-0-14
Degree 22
Conductor 400400
Sign 0.498+0.866i0.498 + 0.866i
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  + (1.94 − 0.990i)3-s + (1.03 − 1.98i)5-s + (1.48 − 1.48i)7-s + (1.03 − 1.42i)9-s + (−0.172 − 0.236i)11-s + (−4.46 + 0.706i)13-s + (0.0547 − 4.88i)15-s + (−1.99 + 3.92i)17-s + (0.873 + 2.68i)19-s + (1.41 − 4.35i)21-s + (3.88 + 0.615i)23-s + (−2.84 − 4.11i)25-s + (−0.421 + 2.66i)27-s + (−5.50 − 1.78i)29-s + (9.13 − 2.96i)31-s + ⋯
L(s)  = 1  + (1.12 − 0.572i)3-s + (0.463 − 0.885i)5-s + (0.560 − 0.560i)7-s + (0.345 − 0.475i)9-s + (−0.0518 − 0.0714i)11-s + (−1.23 + 0.196i)13-s + (0.0141 − 1.26i)15-s + (−0.484 + 0.950i)17-s + (0.200 + 0.616i)19-s + (0.308 − 0.950i)21-s + (0.810 + 0.128i)23-s + (−0.569 − 0.822i)25-s + (−0.0811 + 0.512i)27-s + (−1.02 − 0.331i)29-s + (1.64 − 0.533i)31-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((0.498+0.866i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+1/2) L(s)=((0.498+0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 0.498+0.866i0.498 + 0.866i
Analytic conductor: 3.194013.19401
Root analytic conductor: 1.787181.78718
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ400(127,⋅)\chi_{400} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :1/2), 0.498+0.866i)(2,\ 400,\ (\ :1/2),\ 0.498 + 0.866i)

Particular Values

L(1)L(1) ≈\approx 1.78631−1.03312i1.78631 - 1.03312i
L(12)L(\frac12) ≈\approx 1.78631−1.03312i1.78631 - 1.03312i
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+(−1.03+1.98i)T 1 + (-1.03 + 1.98i)T
good3 1+(−1.94+0.990i)T+(1.76−2.42i)T2 1 + (-1.94 + 0.990i)T + (1.76 - 2.42i)T^{2}
7 1+(−1.48+1.48i)T−7iT2 1 + (-1.48 + 1.48i)T - 7iT^{2}
11 1+(0.172+0.236i)T+(−3.39+10.4i)T2 1 + (0.172 + 0.236i)T + (-3.39 + 10.4i)T^{2}
13 1+(4.46−0.706i)T+(12.3−4.01i)T2 1 + (4.46 - 0.706i)T + (12.3 - 4.01i)T^{2}
17 1+(1.99−3.92i)T+(−9.99−13.7i)T2 1 + (1.99 - 3.92i)T + (-9.99 - 13.7i)T^{2}
19 1+(−0.873−2.68i)T+(−15.3+11.1i)T2 1 + (-0.873 - 2.68i)T + (-15.3 + 11.1i)T^{2}
23 1+(−3.88−0.615i)T+(21.8+7.10i)T2 1 + (-3.88 - 0.615i)T + (21.8 + 7.10i)T^{2}
29 1+(5.50+1.78i)T+(23.4+17.0i)T2 1 + (5.50 + 1.78i)T + (23.4 + 17.0i)T^{2}
31 1+(−9.13+2.96i)T+(25.0−18.2i)T2 1 + (-9.13 + 2.96i)T + (25.0 - 18.2i)T^{2}
37 1+(−0.750−4.73i)T+(−35.1+11.4i)T2 1 + (-0.750 - 4.73i)T + (-35.1 + 11.4i)T^{2}
41 1+(−5.38−3.91i)T+(12.6+38.9i)T2 1 + (-5.38 - 3.91i)T + (12.6 + 38.9i)T^{2}
43 1+(−5.39−5.39i)T+43iT2 1 + (-5.39 - 5.39i)T + 43iT^{2}
47 1+(3.24+6.36i)T+(−27.6+38.0i)T2 1 + (3.24 + 6.36i)T + (-27.6 + 38.0i)T^{2}
53 1+(3.61+7.10i)T+(−31.1+42.8i)T2 1 + (3.61 + 7.10i)T + (-31.1 + 42.8i)T^{2}
59 1+(2.43+1.76i)T+(18.2+56.1i)T2 1 + (2.43 + 1.76i)T + (18.2 + 56.1i)T^{2}
61 1+(−9.58+6.96i)T+(18.8−58.0i)T2 1 + (-9.58 + 6.96i)T + (18.8 - 58.0i)T^{2}
67 1+(8.32+4.24i)T+(39.3+54.2i)T2 1 + (8.32 + 4.24i)T + (39.3 + 54.2i)T^{2}
71 1+(0.108+0.0353i)T+(57.4+41.7i)T2 1 + (0.108 + 0.0353i)T + (57.4 + 41.7i)T^{2}
73 1+(0.879−5.55i)T+(−69.4−22.5i)T2 1 + (0.879 - 5.55i)T + (-69.4 - 22.5i)T^{2}
79 1+(4.32−13.3i)T+(−63.9−46.4i)T2 1 + (4.32 - 13.3i)T + (-63.9 - 46.4i)T^{2}
83 1+(−7.48+14.6i)T+(−48.7−67.1i)T2 1 + (-7.48 + 14.6i)T + (-48.7 - 67.1i)T^{2}
89 1+(−6.38−8.78i)T+(−27.5+84.6i)T2 1 + (-6.38 - 8.78i)T + (-27.5 + 84.6i)T^{2}
97 1+(2.01−1.02i)T+(57.0−78.4i)T2 1 + (2.01 - 1.02i)T + (57.0 - 78.4i)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

−11.15559549443059610619375844530, −9.921201572946360496110250805358, −9.238334640022060600344108242381, −8.136653232046955692583912235982, −7.81182196089365055904955238005, −6.53629089943253889355035058539, −5.16001873475884813354336474785, −4.14500829349107932872878517577, −2.55797224378132654123185607044, −1.45474549360290967106740054848, 2.41156688504720061859795418932, 2.92488824550610112115584873825, 4.45969529115110917216741059412, 5.51687265303169344076690698053, 6.95755356152821835209577006623, 7.72680331092961605302770580838, 8.984530295924776770520380091692, 9.407815006979187143001539027649, 10.37595843252524476468312381895, 11.27601493282348539310886309512

Graph of the ZZ-function along the critical line