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Properties

Label 2-650-13.3-c1-0-1
Degree 22
Conductor 650650
Sign −0.755−0.655i-0.755 - 0.655i
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  + (−0.5 − 0.866i)2-s + (1.36 + 2.36i)3-s + (−0.499 + 0.866i)4-s + (1.36 − 2.36i)6-s + (−2.23 + 3.86i)7-s + 0.999·8-s + (−2.23 + 3.86i)9-s + (−0.866 − 1.5i)11-s − 2.73·12-s + (−3.59 + 0.232i)13-s + 4.46·14-s + (−0.5 − 0.866i)16-s + (2.36 − 4.09i)17-s + 4.46·18-s + (−3.59 + 6.23i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.788 + 1.36i)3-s + (−0.249 + 0.433i)4-s + (0.557 − 0.965i)6-s + (−0.843 + 1.46i)7-s + 0.353·8-s + (−0.744 + 1.28i)9-s + (−0.261 − 0.452i)11-s − 0.788·12-s + (−0.997 + 0.0643i)13-s + 1.19·14-s + (−0.125 − 0.216i)16-s + (0.573 − 0.993i)17-s + 1.05·18-s + (−0.825 + 1.42i)19-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((−0.755−0.655i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+1/2) L(s)=((−0.755−0.655i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: −0.755−0.655i-0.755 - 0.655i
Analytic conductor: 5.190275.19027
Root analytic conductor: 2.278212.27821
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ650(601,⋅)\chi_{650} (601, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :1/2), −0.755−0.655i)(2,\ 650,\ (\ :1/2),\ -0.755 - 0.655i)

Particular Values

L(1)L(1) ≈\approx 0.343431+0.919954i0.343431 + 0.919954i
L(12)L(\frac12) ≈\approx 0.343431+0.919954i0.343431 + 0.919954i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
5 1 1
13 1+(3.59−0.232i)T 1 + (3.59 - 0.232i)T
good3 1+(−1.36−2.36i)T+(−1.5+2.59i)T2 1 + (-1.36 - 2.36i)T + (-1.5 + 2.59i)T^{2}
7 1+(2.23−3.86i)T+(−3.5−6.06i)T2 1 + (2.23 - 3.86i)T + (-3.5 - 6.06i)T^{2}
11 1+(0.866+1.5i)T+(−5.5+9.52i)T2 1 + (0.866 + 1.5i)T + (-5.5 + 9.52i)T^{2}
17 1+(−2.36+4.09i)T+(−8.5−14.7i)T2 1 + (-2.36 + 4.09i)T + (-8.5 - 14.7i)T^{2}
19 1+(3.59−6.23i)T+(−9.5−16.4i)T2 1 + (3.59 - 6.23i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.73+3i)T+(−11.5+19.9i)T2 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.26+2.19i)T+(−14.5+25.1i)T2 1 + (1.26 + 2.19i)T + (-14.5 + 25.1i)T^{2}
31 1−6.73T+31T2 1 - 6.73T + 31T^{2}
37 1+(−0.598−1.03i)T+(−18.5+32.0i)T2 1 + (-0.598 - 1.03i)T + (-18.5 + 32.0i)T^{2}
41 1+(−3.46−6i)T+(−20.5+35.5i)T2 1 + (-3.46 - 6i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.46−4.26i)T+(−21.5−37.2i)T2 1 + (2.46 - 4.26i)T + (-21.5 - 37.2i)T^{2}
47 1+0.464T+47T2 1 + 0.464T + 47T^{2}
53 1+1.73T+53T2 1 + 1.73T + 53T^{2}
59 1+(4.73−8.19i)T+(−29.5−51.0i)T2 1 + (4.73 - 8.19i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.63−2.83i)T+(−30.5−52.8i)T2 1 + (1.63 - 2.83i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.19−10.7i)T+(−33.5+58.0i)T2 1 + (-6.19 - 10.7i)T + (-33.5 + 58.0i)T^{2}
71 1+(4.26−7.39i)T+(−35.5−61.4i)T2 1 + (4.26 - 7.39i)T + (-35.5 - 61.4i)T^{2}
73 1+0.732T+73T2 1 + 0.732T + 73T^{2}
79 1−6.73T+79T2 1 - 6.73T + 79T^{2}
83 1−5.66T+83T2 1 - 5.66T + 83T^{2}
89 1+(−4.5−7.79i)T+(−44.5+77.0i)T2 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2}
97 1+(4.36−7.56i)T+(−48.5−84.0i)T2 1 + (4.36 - 7.56i)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

−10.49144266222012004097235970136, −9.831856307409032872645184183068, −9.440601548266922267162195209131, −8.564060722365837359251142996049, −7.927577955596429008542905619123, −6.24259725220808924826055967879, −5.18643170429196532853531298602, −4.15119722399539605534293757818, −2.97752656888277948434916468307, −2.51531783145462083831709665469, 0.51507192133846132755210390980, 2.01187548205777055911372734968, 3.36452384216245960067513866771, 4.66978377580300877156563387280, 6.24572548106155353088545205998, 6.94836918691122683851685924598, 7.47772488668740152810380700386, 8.137126720931856407990207247855, 9.215353587645191176838525156343, 10.03283118698848312989550640382

Graph of the ZZ-function along the critical line