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Properties

Label 2-650-5.4-c3-0-27
Degree 22
Conductor 650650
Sign 0.894−0.447i0.894 - 0.447i
Analytic cond. 38.351238.3512
Root an. cond. 6.192836.19283
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  + 2i·2-s − 1.62i·3-s − 4·4-s + 3.24·6-s + 2.62i·7-s − 8i·8-s + 24.3·9-s − 51.1·11-s + 6.49i·12-s − 13i·13-s − 5.24·14-s + 16·16-s + 4.36i·17-s + 48.7i·18-s + 47.4·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.312i·3-s − 0.5·4-s + 0.220·6-s + 0.141i·7-s − 0.353i·8-s + 0.902·9-s − 1.40·11-s + 0.156i·12-s − 0.277i·13-s − 0.100·14-s + 0.250·16-s + 0.0622i·17-s + 0.638i·18-s + 0.573·19-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+3/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: 0.894−0.447i0.894 - 0.447i
Analytic conductor: 38.351238.3512
Root analytic conductor: 6.192836.19283
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ650(599,⋅)\chi_{650} (599, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :3/2), 0.894−0.447i)(2,\ 650,\ (\ :3/2),\ 0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 1.7883239631.788323963
L(12)L(\frac12) ≈\approx 1.7883239631.788323963
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 1−2iT 1 - 2iT
5 1 1
13 1+13iT 1 + 13iT
good3 1+1.62iT−27T2 1 + 1.62iT - 27T^{2}
7 1−2.62iT−343T2 1 - 2.62iT - 343T^{2}
11 1+51.1T+1.33e3T2 1 + 51.1T + 1.33e3T^{2}
17 1−4.36iT−4.91e3T2 1 - 4.36iT - 4.91e3T^{2}
19 1−47.4T+6.85e3T2 1 - 47.4T + 6.85e3T^{2}
23 1−91.5iT−1.21e4T2 1 - 91.5iT - 1.21e4T^{2}
29 1−139.T+2.43e4T2 1 - 139.T + 2.43e4T^{2}
31 1−31.1T+2.97e4T2 1 - 31.1T + 2.97e4T^{2}
37 1+377.iT−5.06e4T2 1 + 377. iT - 5.06e4T^{2}
41 1+7.71T+6.89e4T2 1 + 7.71T + 6.89e4T^{2}
43 1+75.5iT−7.95e4T2 1 + 75.5iT - 7.95e4T^{2}
47 1−186.iT−1.03e5T2 1 - 186. iT - 1.03e5T^{2}
53 1+236.iT−1.48e5T2 1 + 236. iT - 1.48e5T^{2}
59 1−4.36T+2.05e5T2 1 - 4.36T + 2.05e5T^{2}
61 1−380.T+2.26e5T2 1 - 380.T + 2.26e5T^{2}
67 1−98.2iT−3.00e5T2 1 - 98.2iT - 3.00e5T^{2}
71 1−1.16e3T+3.57e5T2 1 - 1.16e3T + 3.57e5T^{2}
73 1−404.iT−3.89e5T2 1 - 404. iT - 3.89e5T^{2}
79 1−856.T+4.93e5T2 1 - 856.T + 4.93e5T^{2}
83 1−920.iT−5.71e5T2 1 - 920. iT - 5.71e5T^{2}
89 1−1.52e3T+7.04e5T2 1 - 1.52e3T + 7.04e5T^{2}
97 1+960.iT−9.12e5T2 1 + 960. iT - 9.12e5T^{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

−10.11260102476124476804925987694, −9.319338338000401015657113912916, −8.158542131109715790041623330803, −7.59159891177575733982919487694, −6.80618862138402083258021255408, −5.66334557253068368440901373465, −4.95734142805661577683190918634, −3.72395545287353145770519826938, −2.33149936289840854596807597285, −0.75220340419365490380966276786, 0.830669908655800126291687768571, 2.25631649111312587671975089838, 3.33159046459911736920967532245, 4.52452948077747458748486351323, 5.14699670409903873733809223291, 6.52883180508908881034571198945, 7.60572993251976460974912084798, 8.431060422507145641270022636387, 9.504625471068891837142373137078, 10.26949456076721477479630390336

Graph of the ZZ-function along the critical line