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Properties

Label 2-20e2-100.27-c1-0-4
Degree 22
Conductor 400400
Sign −0.0673−0.997i-0.0673 - 0.997i
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  + (−2.97 + 1.51i)3-s + (1.69 + 1.45i)5-s + (1.80 − 1.80i)7-s + (4.80 − 6.61i)9-s + (2.46 + 3.39i)11-s + (0.731 − 0.115i)13-s + (−7.26 − 1.76i)15-s + (−3.09 + 6.08i)17-s + (0.285 + 0.877i)19-s + (−2.63 + 8.10i)21-s + (−3.66 − 0.579i)23-s + (0.753 + 4.94i)25-s + (−2.70 + 17.0i)27-s + (−2.48 − 0.808i)29-s + (5.85 − 1.90i)31-s + ⋯
L(s)  = 1  + (−1.71 + 0.876i)3-s + (0.758 + 0.651i)5-s + (0.681 − 0.681i)7-s + (1.60 − 2.20i)9-s + (0.743 + 1.02i)11-s + (0.202 − 0.0321i)13-s + (−1.87 − 0.456i)15-s + (−0.751 + 1.47i)17-s + (0.0653 + 0.201i)19-s + (−0.574 + 1.76i)21-s + (−0.763 − 0.120i)23-s + (0.150 + 0.988i)25-s + (−0.520 + 3.28i)27-s + (−0.462 − 0.150i)29-s + (1.05 − 0.341i)31-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((−0.0673−0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0673 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+1/2) L(s)=((−0.0673−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0673 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: −0.0673−0.997i-0.0673 - 0.997i
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), −0.0673−0.997i)(2,\ 400,\ (\ :1/2),\ -0.0673 - 0.997i)

Particular Values

L(1)L(1) ≈\approx 0.640902+0.685645i0.640902 + 0.685645i
L(12)L(\frac12) ≈\approx 0.640902+0.685645i0.640902 + 0.685645i
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+(−1.69−1.45i)T 1 + (-1.69 - 1.45i)T
good3 1+(2.97−1.51i)T+(1.76−2.42i)T2 1 + (2.97 - 1.51i)T + (1.76 - 2.42i)T^{2}
7 1+(−1.80+1.80i)T−7iT2 1 + (-1.80 + 1.80i)T - 7iT^{2}
11 1+(−2.46−3.39i)T+(−3.39+10.4i)T2 1 + (-2.46 - 3.39i)T + (-3.39 + 10.4i)T^{2}
13 1+(−0.731+0.115i)T+(12.3−4.01i)T2 1 + (-0.731 + 0.115i)T + (12.3 - 4.01i)T^{2}
17 1+(3.09−6.08i)T+(−9.99−13.7i)T2 1 + (3.09 - 6.08i)T + (-9.99 - 13.7i)T^{2}
19 1+(−0.285−0.877i)T+(−15.3+11.1i)T2 1 + (-0.285 - 0.877i)T + (-15.3 + 11.1i)T^{2}
23 1+(3.66+0.579i)T+(21.8+7.10i)T2 1 + (3.66 + 0.579i)T + (21.8 + 7.10i)T^{2}
29 1+(2.48+0.808i)T+(23.4+17.0i)T2 1 + (2.48 + 0.808i)T + (23.4 + 17.0i)T^{2}
31 1+(−5.85+1.90i)T+(25.0−18.2i)T2 1 + (-5.85 + 1.90i)T + (25.0 - 18.2i)T^{2}
37 1+(0.443+2.79i)T+(−35.1+11.4i)T2 1 + (0.443 + 2.79i)T + (-35.1 + 11.4i)T^{2}
41 1+(−2.09−1.52i)T+(12.6+38.9i)T2 1 + (-2.09 - 1.52i)T + (12.6 + 38.9i)T^{2}
43 1+(−1.85−1.85i)T+43iT2 1 + (-1.85 - 1.85i)T + 43iT^{2}
47 1+(−3.13−6.15i)T+(−27.6+38.0i)T2 1 + (-3.13 - 6.15i)T + (-27.6 + 38.0i)T^{2}
53 1+(−0.786−1.54i)T+(−31.1+42.8i)T2 1 + (-0.786 - 1.54i)T + (-31.1 + 42.8i)T^{2}
59 1+(−1.10−0.803i)T+(18.2+56.1i)T2 1 + (-1.10 - 0.803i)T + (18.2 + 56.1i)T^{2}
61 1+(6.13−4.45i)T+(18.8−58.0i)T2 1 + (6.13 - 4.45i)T + (18.8 - 58.0i)T^{2}
67 1+(8.25+4.20i)T+(39.3+54.2i)T2 1 + (8.25 + 4.20i)T + (39.3 + 54.2i)T^{2}
71 1+(−5.26−1.71i)T+(57.4+41.7i)T2 1 + (-5.26 - 1.71i)T + (57.4 + 41.7i)T^{2}
73 1+(−0.380+2.40i)T+(−69.4−22.5i)T2 1 + (-0.380 + 2.40i)T + (-69.4 - 22.5i)T^{2}
79 1+(−1.89+5.82i)T+(−63.9−46.4i)T2 1 + (-1.89 + 5.82i)T + (-63.9 - 46.4i)T^{2}
83 1+(1.10−2.16i)T+(−48.7−67.1i)T2 1 + (1.10 - 2.16i)T + (-48.7 - 67.1i)T^{2}
89 1+(5.88+8.09i)T+(−27.5+84.6i)T2 1 + (5.88 + 8.09i)T + (-27.5 + 84.6i)T^{2}
97 1+(−7.27+3.70i)T+(57.0−78.4i)T2 1 + (-7.27 + 3.70i)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

−11.25624733760191228408402063806, −10.60349567382711583631959929928, −10.09078369223113986770752214339, −9.202202339946898203303851038783, −7.47357624474988366148228234696, −6.39561029723586226346977303354, −5.93627030067282796501048034115, −4.58812780978632242961456764465, −4.00634676897108344556292119752, −1.56773512573196279765147817836, 0.845275558661012419551379080891, 2.09955271388581715263552075923, 4.63768812113616285565132745776, 5.41767974574921160181859832470, 6.11080705320614540937154628422, 6.95194931495880922631541077222, 8.234260705095642197344623692431, 9.190195327222392721681192214553, 10.39246067915401064771416873574, 11.46611459429946702837601117964

Graph of the ZZ-function along the critical line