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Properties

Label 2-20e2-20.3-c3-0-16
Degree 22
Conductor 400400
Sign 0.880+0.473i0.880 + 0.473i
Analytic cond. 23.600723.6007
Root an. cond. 4.858064.85806
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.99 + 2.99i)3-s + (6.68 + 6.68i)7-s + 9.09i·9-s − 63.1i·11-s + (−28.2 − 28.2i)13-s + (−30.3 + 30.3i)17-s + 22.2·19-s − 40.0·21-s + (86.5 − 86.5i)23-s + (−107. − 107. i)27-s + 188. i·29-s + 71.2i·31-s + (188. + 188. i)33-s + (217. − 217. i)37-s + 168.·39-s + ⋯
L(s)  = 1  + (−0.575 + 0.575i)3-s + (0.361 + 0.361i)7-s + 0.336i·9-s − 1.73i·11-s + (−0.602 − 0.602i)13-s + (−0.433 + 0.433i)17-s + 0.268·19-s − 0.415·21-s + (0.784 − 0.784i)23-s + (−0.769 − 0.769i)27-s + 1.20i·29-s + 0.413i·31-s + (0.996 + 0.996i)33-s + (0.964 − 0.964i)37-s + 0.693·39-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((0.880+0.473i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+3/2) L(s)=((0.880+0.473i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 0.880+0.473i0.880 + 0.473i
Analytic conductor: 23.600723.6007
Root analytic conductor: 4.858064.85806
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ400(143,⋅)\chi_{400} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :3/2), 0.880+0.473i)(2,\ 400,\ (\ :3/2),\ 0.880 + 0.473i)

Particular Values

L(2)L(2) ≈\approx 1.3180636331.318063633
L(12)L(\frac12) ≈\approx 1.3180636331.318063633
L(52)L(\frac{5}{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+(2.99−2.99i)T−27iT2 1 + (2.99 - 2.99i)T - 27iT^{2}
7 1+(−6.68−6.68i)T+343iT2 1 + (-6.68 - 6.68i)T + 343iT^{2}
11 1+63.1iT−1.33e3T2 1 + 63.1iT - 1.33e3T^{2}
13 1+(28.2+28.2i)T+2.19e3iT2 1 + (28.2 + 28.2i)T + 2.19e3iT^{2}
17 1+(30.3−30.3i)T−4.91e3iT2 1 + (30.3 - 30.3i)T - 4.91e3iT^{2}
19 1−22.2T+6.85e3T2 1 - 22.2T + 6.85e3T^{2}
23 1+(−86.5+86.5i)T−1.21e4iT2 1 + (-86.5 + 86.5i)T - 1.21e4iT^{2}
29 1−188.iT−2.43e4T2 1 - 188. iT - 2.43e4T^{2}
31 1−71.2iT−2.97e4T2 1 - 71.2iT - 2.97e4T^{2}
37 1+(−217.+217.i)T−5.06e4iT2 1 + (-217. + 217. i)T - 5.06e4iT^{2}
41 1+4.57T+6.89e4T2 1 + 4.57T + 6.89e4T^{2}
43 1+(−392.+392.i)T−7.95e4iT2 1 + (-392. + 392. i)T - 7.95e4iT^{2}
47 1+(280.+280.i)T+1.03e5iT2 1 + (280. + 280. i)T + 1.03e5iT^{2}
53 1+(−82.3−82.3i)T+1.48e5iT2 1 + (-82.3 - 82.3i)T + 1.48e5iT^{2}
59 1−794.T+2.05e5T2 1 - 794.T + 2.05e5T^{2}
61 1−714.T+2.26e5T2 1 - 714.T + 2.26e5T^{2}
67 1+(−373.−373.i)T+3.00e5iT2 1 + (-373. - 373. i)T + 3.00e5iT^{2}
71 1+528.iT−3.57e5T2 1 + 528. iT - 3.57e5T^{2}
73 1+(−531.−531.i)T+3.89e5iT2 1 + (-531. - 531. i)T + 3.89e5iT^{2}
79 1+349.T+4.93e5T2 1 + 349.T + 4.93e5T^{2}
83 1+(−112.+112.i)T−5.71e5iT2 1 + (-112. + 112. i)T - 5.71e5iT^{2}
89 1−424.iT−7.04e5T2 1 - 424. iT - 7.04e5T^{2}
97 1+(501.−501.i)T−9.12e5iT2 1 + (501. - 501. i)T - 9.12e5iT^{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

−10.92469615219343023655939263719, −10.10854258770323050739733462813, −8.832745098504614378296036253465, −8.229777221881297331462095969995, −6.93335114427589800256865515918, −5.62524950371248648185549050737, −5.19278064360904661654743561138, −3.83094657255593621080382250453, −2.50270169543341845881814358070, −0.57525860402511737446073501773, 1.07327536206683353925103534015, 2.36630351252272570435169948379, 4.18138792932687291488371482227, 5.02776292652152002668864732555, 6.36502129284057763690112419594, 7.15337756203982981649716660407, 7.80106818225563519239630346585, 9.430698802162173229527846855854, 9.788146008405428812842714074200, 11.28029510031533832678395674715

Graph of the ZZ-function along the critical line