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Properties

Label 2-20e2-5.2-c2-0-1
Degree 22
Conductor 400400
Sign −0.850−0.525i-0.850 - 0.525i
Analytic cond. 10.899210.8992
Root an. cond. 3.301393.30139
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)3-s + (−7 − 7i)7-s + 7i·9-s − 10·11-s + (−9 + 9i)13-s + (−1 − i)17-s + 8i·19-s − 14·21-s + (−23 + 23i)23-s + (16 + 16i)27-s − 8i·29-s + 14·31-s + (−10 + 10i)33-s + (−33 − 33i)37-s + 18i·39-s + ⋯
L(s)  = 1  + (0.333 − 0.333i)3-s + (−1 − i)7-s + 0.777i·9-s − 0.909·11-s + (−0.692 + 0.692i)13-s + (−0.0588 − 0.0588i)17-s + 0.421i·19-s − 0.666·21-s + (−1 + i)23-s + (0.592 + 0.592i)27-s − 0.275i·29-s + 0.451·31-s + (−0.303 + 0.303i)33-s + (−0.891 − 0.891i)37-s + 0.461i·39-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((−0.850−0.525i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+1) L(s)=((−0.850−0.525i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: −0.850−0.525i-0.850 - 0.525i
Analytic conductor: 10.899210.8992
Root analytic conductor: 3.301393.30139
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ400(257,⋅)\chi_{400} (257, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :1), −0.850−0.525i)(2,\ 400,\ (\ :1),\ -0.850 - 0.525i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0512526+0.180416i0.0512526 + 0.180416i
L(12)L(\frac12) ≈\approx 0.0512526+0.180416i0.0512526 + 0.180416i
L(2)L(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+(−1+i)T−9iT2 1 + (-1 + i)T - 9iT^{2}
7 1+(7+7i)T+49iT2 1 + (7 + 7i)T + 49iT^{2}
11 1+10T+121T2 1 + 10T + 121T^{2}
13 1+(9−9i)T−169iT2 1 + (9 - 9i)T - 169iT^{2}
17 1+(1+i)T+289iT2 1 + (1 + i)T + 289iT^{2}
19 1−8iT−361T2 1 - 8iT - 361T^{2}
23 1+(23−23i)T−529iT2 1 + (23 - 23i)T - 529iT^{2}
29 1+8iT−841T2 1 + 8iT - 841T^{2}
31 1−14T+961T2 1 - 14T + 961T^{2}
37 1+(33+33i)T+1.36e3iT2 1 + (33 + 33i)T + 1.36e3iT^{2}
41 1+14T+1.68e3T2 1 + 14T + 1.68e3T^{2}
43 1+(15−15i)T−1.84e3iT2 1 + (15 - 15i)T - 1.84e3iT^{2}
47 1+(39+39i)T+2.20e3iT2 1 + (39 + 39i)T + 2.20e3iT^{2}
53 1+(−7+7i)T−2.80e3iT2 1 + (-7 + 7i)T - 2.80e3iT^{2}
59 1−56iT−3.48e3T2 1 - 56iT - 3.48e3T^{2}
61 1−42T+3.72e3T2 1 - 42T + 3.72e3T^{2}
67 1+(7+7i)T+4.48e3iT2 1 + (7 + 7i)T + 4.48e3iT^{2}
71 1+98T+5.04e3T2 1 + 98T + 5.04e3T^{2}
73 1+(49−49i)T−5.32e3iT2 1 + (49 - 49i)T - 5.32e3iT^{2}
79 1+96iT−6.24e3T2 1 + 96iT - 6.24e3T^{2}
83 1+(63−63i)T−6.88e3iT2 1 + (63 - 63i)T - 6.88e3iT^{2}
89 1+112iT−7.92e3T2 1 + 112iT - 7.92e3T^{2}
97 1+(33+33i)T+9.40e3iT2 1 + (33 + 33i)T + 9.40e3iT^{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

−11.41499513159046605609880188291, −10.22727968251262811586009858664, −9.924881399325737445108897709911, −8.572085299272496551984124415464, −7.54623056790744008073582091156, −7.03152782655453350323850298444, −5.72744098947145072525660119313, −4.48457765401652588862314413602, −3.26555131103176463657106291736, −1.96140755417840218347593722656, 0.07153432332835254383280210600, 2.53956892720900427113331112727, 3.30990526953465270386917161561, 4.80020209566613135711326567837, 5.90458906093560965323063219196, 6.77842899970267672950284916551, 8.120787698760519666875659126330, 8.896855359243206894741192457438, 9.848243468833479780588358365579, 10.34075574639242689201889643153

Graph of the ZZ-function along the critical line