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Properties

Label 2-405-135.122-c1-0-12
Degree 22
Conductor 405405
Sign 0.775+0.630i0.775 + 0.630i
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
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  + (0.807 + 0.565i)2-s + (−0.351 − 0.966i)4-s + (2.04 − 0.911i)5-s + (−0.275 − 0.590i)7-s + (0.772 − 2.88i)8-s + (2.16 + 0.418i)10-s + (−0.890 − 1.06i)11-s + (−2.93 − 4.19i)13-s + (0.111 − 0.632i)14-s + (0.677 − 0.568i)16-s + (1.18 + 4.41i)17-s + (0.00652 − 0.00376i)19-s + (−1.59 − 1.65i)20-s + (−0.118 − 1.36i)22-s + (6.70 + 3.12i)23-s + ⋯
L(s)  = 1  + (0.570 + 0.399i)2-s + (−0.175 − 0.483i)4-s + (0.913 − 0.407i)5-s + (−0.104 − 0.223i)7-s + (0.273 − 1.01i)8-s + (0.684 + 0.132i)10-s + (−0.268 − 0.319i)11-s + (−0.814 − 1.16i)13-s + (0.0298 − 0.169i)14-s + (0.169 − 0.142i)16-s + (0.286 + 1.06i)17-s + (0.00149 − 0.000864i)19-s + (−0.357 − 0.369i)20-s + (−0.0253 − 0.289i)22-s + (1.39 + 0.651i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((0.775+0.630i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((0.775+0.630i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: 0.775+0.630i0.775 + 0.630i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(152,⋅)\chi_{405} (152, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), 0.775+0.630i)(2,\ 405,\ (\ :1/2),\ 0.775 + 0.630i)

Particular Values

L(1)L(1) ≈\approx 1.80543−0.641221i1.80543 - 0.641221i
L(12)L(\frac12) ≈\approx 1.80543−0.641221i1.80543 - 0.641221i
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)
bad3 1 1
5 1+(−2.04+0.911i)T 1 + (-2.04 + 0.911i)T
good2 1+(−0.807−0.565i)T+(0.684+1.87i)T2 1 + (-0.807 - 0.565i)T + (0.684 + 1.87i)T^{2}
7 1+(0.275+0.590i)T+(−4.49+5.36i)T2 1 + (0.275 + 0.590i)T + (-4.49 + 5.36i)T^{2}
11 1+(0.890+1.06i)T+(−1.91+10.8i)T2 1 + (0.890 + 1.06i)T + (-1.91 + 10.8i)T^{2}
13 1+(2.93+4.19i)T+(−4.44+12.2i)T2 1 + (2.93 + 4.19i)T + (-4.44 + 12.2i)T^{2}
17 1+(−1.18−4.41i)T+(−14.7+8.5i)T2 1 + (-1.18 - 4.41i)T + (-14.7 + 8.5i)T^{2}
19 1+(−0.00652+0.00376i)T+(9.5−16.4i)T2 1 + (-0.00652 + 0.00376i)T + (9.5 - 16.4i)T^{2}
23 1+(−6.70−3.12i)T+(14.7+17.6i)T2 1 + (-6.70 - 3.12i)T + (14.7 + 17.6i)T^{2}
29 1+(−0.586−3.32i)T+(−27.2+9.91i)T2 1 + (-0.586 - 3.32i)T + (-27.2 + 9.91i)T^{2}
31 1+(−3.73+1.35i)T+(23.7−19.9i)T2 1 + (-3.73 + 1.35i)T + (23.7 - 19.9i)T^{2}
37 1+(−1.60+0.430i)T+(32.0−18.5i)T2 1 + (-1.60 + 0.430i)T + (32.0 - 18.5i)T^{2}
41 1+(−1.90−0.335i)T+(38.5+14.0i)T2 1 + (-1.90 - 0.335i)T + (38.5 + 14.0i)T^{2}
43 1+(0.929−10.6i)T+(−42.3−7.46i)T2 1 + (0.929 - 10.6i)T + (-42.3 - 7.46i)T^{2}
47 1+(6.61−3.08i)T+(30.2−36.0i)T2 1 + (6.61 - 3.08i)T + (30.2 - 36.0i)T^{2}
53 1+(−1.18−1.18i)T+53iT2 1 + (-1.18 - 1.18i)T + 53iT^{2}
59 1+(−7.77−6.52i)T+(10.2+58.1i)T2 1 + (-7.77 - 6.52i)T + (10.2 + 58.1i)T^{2}
61 1+(9.02+3.28i)T+(46.7+39.2i)T2 1 + (9.02 + 3.28i)T + (46.7 + 39.2i)T^{2}
67 1+(12.3−8.62i)T+(22.9−62.9i)T2 1 + (12.3 - 8.62i)T + (22.9 - 62.9i)T^{2}
71 1+(5.41+3.12i)T+(35.5+61.4i)T2 1 + (5.41 + 3.12i)T + (35.5 + 61.4i)T^{2}
73 1+(−12.1−3.25i)T+(63.2+36.5i)T2 1 + (-12.1 - 3.25i)T + (63.2 + 36.5i)T^{2}
79 1+(0.782−0.137i)T+(74.2−27.0i)T2 1 + (0.782 - 0.137i)T + (74.2 - 27.0i)T^{2}
83 1+(5.32−7.59i)T+(−28.3−77.9i)T2 1 + (5.32 - 7.59i)T + (-28.3 - 77.9i)T^{2}
89 1+(5.89+10.2i)T+(−44.5+77.0i)T2 1 + (5.89 + 10.2i)T + (-44.5 + 77.0i)T^{2}
97 1+(2.18+0.190i)T+(95.5+16.8i)T2 1 + (2.18 + 0.190i)T + (95.5 + 16.8i)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

−10.92407568586431660469202045297, −10.15815700553343106979694071640, −9.536589640838619372751374684035, −8.398646395438730572100019533808, −7.21591413529217253580804091607, −6.09458651965138779794414717970, −5.43366493442251769542124773850, −4.58382788041239992241516300878, −3.02776310722934582923420773692, −1.17921448290620566959215892189, 2.20024352431318745740436581391, 3.00943220754072508205102632688, 4.54336634275867436610717101716, 5.28136498349557179441767659792, 6.66316980370285431611096368681, 7.46832072610149796713753973754, 8.835609351522906087957626827584, 9.532672145774340239691343654926, 10.53789001842868626659561301315, 11.56738997697051011606428115284

Graph of the ZZ-function along the critical line