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Properties

Label 2-20e2-20.3-c3-0-20
Degree 22
Conductor 400400
Sign 0.559+0.828i0.559 + 0.828i
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  + (2.35 − 2.35i)3-s + (−0.458 − 0.458i)7-s + 15.9i·9-s − 23.0i·11-s + (−1.65 − 1.65i)13-s + (63.9 − 63.9i)17-s + 65.6·19-s − 2.15·21-s + (45.2 − 45.2i)23-s + (100. + 100. i)27-s + 59.8i·29-s − 279. i·31-s + (−54.2 − 54.2i)33-s + (100. − 100. i)37-s − 7.77·39-s + ⋯
L(s)  = 1  + (0.452 − 0.452i)3-s + (−0.0247 − 0.0247i)7-s + 0.590i·9-s − 0.632i·11-s + (−0.0352 − 0.0352i)13-s + (0.912 − 0.912i)17-s + 0.793·19-s − 0.0224·21-s + (0.410 − 0.410i)23-s + (0.719 + 0.719i)27-s + 0.383i·29-s − 1.61i·31-s + (−0.286 − 0.286i)33-s + (0.446 − 0.446i)37-s − 0.0319·39-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((0.559+0.828i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+3/2) L(s)=((0.559+0.828i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 0.559+0.828i0.559 + 0.828i
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), 0.559+0.828i)(2,\ 400,\ (\ :3/2),\ 0.559 + 0.828i)

Particular Values

L(2)L(2) ≈\approx 2.2355530742.235553074
L(12)L(\frac12) ≈\approx 2.2355530742.235553074
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+(−2.35+2.35i)T−27iT2 1 + (-2.35 + 2.35i)T - 27iT^{2}
7 1+(0.458+0.458i)T+343iT2 1 + (0.458 + 0.458i)T + 343iT^{2}
11 1+23.0iT−1.33e3T2 1 + 23.0iT - 1.33e3T^{2}
13 1+(1.65+1.65i)T+2.19e3iT2 1 + (1.65 + 1.65i)T + 2.19e3iT^{2}
17 1+(−63.9+63.9i)T−4.91e3iT2 1 + (-63.9 + 63.9i)T - 4.91e3iT^{2}
19 1−65.6T+6.85e3T2 1 - 65.6T + 6.85e3T^{2}
23 1+(−45.2+45.2i)T−1.21e4iT2 1 + (-45.2 + 45.2i)T - 1.21e4iT^{2}
29 1−59.8iT−2.43e4T2 1 - 59.8iT - 2.43e4T^{2}
31 1+279.iT−2.97e4T2 1 + 279. iT - 2.97e4T^{2}
37 1+(−100.+100.i)T−5.06e4iT2 1 + (-100. + 100. i)T - 5.06e4iT^{2}
41 1+274.T+6.89e4T2 1 + 274.T + 6.89e4T^{2}
43 1+(−263.+263.i)T−7.95e4iT2 1 + (-263. + 263. i)T - 7.95e4iT^{2}
47 1+(−391.−391.i)T+1.03e5iT2 1 + (-391. - 391. i)T + 1.03e5iT^{2}
53 1+(265.+265.i)T+1.48e5iT2 1 + (265. + 265. i)T + 1.48e5iT^{2}
59 1−121.T+2.05e5T2 1 - 121.T + 2.05e5T^{2}
61 1+617.T+2.26e5T2 1 + 617.T + 2.26e5T^{2}
67 1+(−674.−674.i)T+3.00e5iT2 1 + (-674. - 674. i)T + 3.00e5iT^{2}
71 1+829.iT−3.57e5T2 1 + 829. iT - 3.57e5T^{2}
73 1+(−110.−110.i)T+3.89e5iT2 1 + (-110. - 110. i)T + 3.89e5iT^{2}
79 1+62.7T+4.93e5T2 1 + 62.7T + 4.93e5T^{2}
83 1+(−655.+655.i)T−5.71e5iT2 1 + (-655. + 655. i)T - 5.71e5iT^{2}
89 1+92.0iT−7.04e5T2 1 + 92.0iT - 7.04e5T^{2}
97 1+(−618.+618.i)T−9.12e5iT2 1 + (-618. + 618. 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

−10.76470365887660149915174020575, −9.723489159206760435727031702512, −8.810315047559895941338817968406, −7.81111822560352489141426194350, −7.22841418417708601311366298835, −5.88396028805799669750964870066, −4.91820287061574096265222211537, −3.41793898933906880988970144473, −2.35763920598425276161860708298, −0.822267001719919992232849534752, 1.27311687418621771021707025739, 2.95712439847344203077826395208, 3.89342476927717586765195205512, 5.07123682323733986675034214385, 6.24453183449785098142901813432, 7.33209792116143031061366447846, 8.320298101152083539107578376555, 9.306470953877234526860556117402, 9.927880642176014125380611684588, 10.83338566816038143812548960866

Graph of the ZZ-function along the critical line