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Properties

Label 2-650-65.37-c1-0-11
Degree 22
Conductor 650650
Sign 0.879+0.476i0.879 + 0.476i
Analytic cond. 5.190275.19027
Root an. cond. 2.278212.27821
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  + (0.866 − 0.5i)2-s + (1.71 + 0.460i)3-s + (0.499 − 0.866i)4-s + (1.71 − 0.460i)6-s + (0.386 − 0.670i)7-s − 0.999i·8-s + (0.142 + 0.0825i)9-s + (2.36 + 0.634i)11-s + (1.25 − 1.25i)12-s + (3.22 + 1.61i)13-s − 0.773i·14-s + (−0.5 − 0.866i)16-s + (−0.325 − 1.21i)17-s + 0.165·18-s + (−0.0463 − 0.173i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.992 + 0.265i)3-s + (0.249 − 0.433i)4-s + (0.701 − 0.187i)6-s + (0.146 − 0.253i)7-s − 0.353i·8-s + (0.0476 + 0.0275i)9-s + (0.713 + 0.191i)11-s + (0.363 − 0.363i)12-s + (0.894 + 0.447i)13-s − 0.206i·14-s + (−0.125 − 0.216i)16-s + (−0.0790 − 0.295i)17-s + 0.0389·18-s + (−0.0106 − 0.0397i)19-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((0.879+0.476i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+1/2) L(s)=((0.879+0.476i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: 0.879+0.476i0.879 + 0.476i
Analytic conductor: 5.190275.19027
Root analytic conductor: 2.278212.27821
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ650(557,⋅)\chi_{650} (557, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :1/2), 0.879+0.476i)(2,\ 650,\ (\ :1/2),\ 0.879 + 0.476i)

Particular Values

L(1)L(1) ≈\approx 2.86938−0.726925i2.86938 - 0.726925i
L(12)L(\frac12) ≈\approx 2.86938−0.726925i2.86938 - 0.726925i
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)
bad2 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
5 1 1
13 1+(−3.22−1.61i)T 1 + (-3.22 - 1.61i)T
good3 1+(−1.71−0.460i)T+(2.59+1.5i)T2 1 + (-1.71 - 0.460i)T + (2.59 + 1.5i)T^{2}
7 1+(−0.386+0.670i)T+(−3.5−6.06i)T2 1 + (-0.386 + 0.670i)T + (-3.5 - 6.06i)T^{2}
11 1+(−2.36−0.634i)T+(9.52+5.5i)T2 1 + (-2.36 - 0.634i)T + (9.52 + 5.5i)T^{2}
17 1+(0.325+1.21i)T+(−14.7+8.5i)T2 1 + (0.325 + 1.21i)T + (-14.7 + 8.5i)T^{2}
19 1+(0.0463+0.173i)T+(−16.4+9.5i)T2 1 + (0.0463 + 0.173i)T + (-16.4 + 9.5i)T^{2}
23 1+(0.0925−0.345i)T+(−19.9−11.5i)T2 1 + (0.0925 - 0.345i)T + (-19.9 - 11.5i)T^{2}
29 1+(−0.581+0.335i)T+(14.5−25.1i)T2 1 + (-0.581 + 0.335i)T + (14.5 - 25.1i)T^{2}
31 1+(−2.01−2.01i)T+31iT2 1 + (-2.01 - 2.01i)T + 31iT^{2}
37 1+(4.27+7.41i)T+(−18.5+32.0i)T2 1 + (4.27 + 7.41i)T + (-18.5 + 32.0i)T^{2}
41 1+(2.60−9.72i)T+(−35.5−20.5i)T2 1 + (2.60 - 9.72i)T + (-35.5 - 20.5i)T^{2}
43 1+(0.194−0.0521i)T+(37.2−21.5i)T2 1 + (0.194 - 0.0521i)T + (37.2 - 21.5i)T^{2}
47 1+9.78T+47T2 1 + 9.78T + 47T^{2}
53 1+(0.918−0.918i)T−53iT2 1 + (0.918 - 0.918i)T - 53iT^{2}
59 1+(0.742−0.199i)T+(51.0−29.5i)T2 1 + (0.742 - 0.199i)T + (51.0 - 29.5i)T^{2}
61 1+(7.31−12.6i)T+(−30.5−52.8i)T2 1 + (7.31 - 12.6i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.894−0.516i)T+(33.5−58.0i)T2 1 + (0.894 - 0.516i)T + (33.5 - 58.0i)T^{2}
71 1+(−4.58+1.22i)T+(61.4−35.5i)T2 1 + (-4.58 + 1.22i)T + (61.4 - 35.5i)T^{2}
73 1+12.2iT−73T2 1 + 12.2iT - 73T^{2}
79 1−8.67iT−79T2 1 - 8.67iT - 79T^{2}
83 1+8.94T+83T2 1 + 8.94T + 83T^{2}
89 1+(−3.34+12.4i)T+(−77.0−44.5i)T2 1 + (-3.34 + 12.4i)T + (-77.0 - 44.5i)T^{2}
97 1+(−9.71−5.61i)T+(48.5+84.0i)T2 1 + (-9.71 - 5.61i)T + (48.5 + 84.0i)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

−10.52846219679753308948854456688, −9.511280760867449173271735377078, −8.916820891397890833896947883639, −7.988444238908903579113633954432, −6.84560690915142961656309224605, −5.94859995507167822420834938421, −4.59931396675276255858086460395, −3.79014049796353800731780902379, −2.90513798971131687046548768345, −1.55624168399136523899918468595, 1.78475156340784982884983459013, 3.08386555756357171051891180973, 3.85061568400383638643661865273, 5.17094185479308876809962999633, 6.16946089007509571197024387040, 7.04464399032874684142961303169, 8.262356008165738958254477963675, 8.462807567753357011826912594857, 9.529224173297660082533370907458, 10.74213680015641298829070863036

Graph of the ZZ-function along the critical line