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Properties

Label 2-31e2-31.19-c1-0-3
Degree 22
Conductor 961961
Sign −0.999−0.00444i-0.999 - 0.00444i
Analytic cond. 7.673627.67362
Root an. cond. 2.770132.77013
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 + 0.363i)2-s + (1.12 + 0.502i)3-s + (−0.5 − 1.53i)4-s + (−0.5 + 0.866i)5-s + (0.381 + 0.661i)6-s + (−2.83 + 3.14i)7-s + (0.690 − 2.12i)8-s + (−0.985 − 1.09i)9-s + (−0.564 + 0.251i)10-s + (−1.95 − 0.415i)11-s + (0.209 − 1.98i)12-s + (−0.129 − 1.22i)13-s + (−2.56 + 0.544i)14-s + (−1 + 0.726i)15-s + (−1.49 + 1.08i)16-s + (−5.12 + 1.08i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.256i)2-s + (0.651 + 0.290i)3-s + (−0.250 − 0.769i)4-s + (−0.223 + 0.387i)5-s + (0.155 + 0.270i)6-s + (−1.07 + 1.18i)7-s + (0.244 − 0.751i)8-s + (−0.328 − 0.364i)9-s + (−0.178 + 0.0794i)10-s + (−0.589 − 0.125i)11-s + (0.0603 − 0.574i)12-s + (−0.0358 − 0.340i)13-s + (−0.684 + 0.145i)14-s + (−0.258 + 0.187i)15-s + (−0.374 + 0.272i)16-s + (−1.24 + 0.264i)17-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=((−0.999−0.00444i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00444i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=((−0.999−0.00444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00444i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: −0.999−0.00444i-0.999 - 0.00444i
Analytic conductor: 7.673627.67362
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ961(732,⋅)\chi_{961} (732, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 961, ( :1/2), −0.999−0.00444i)(2,\ 961,\ (\ :1/2),\ -0.999 - 0.00444i)

Particular Values

L(1)L(1) ≈\approx 0.000743602+0.334756i0.000743602 + 0.334756i
L(12)L(\frac12) ≈\approx 0.000743602+0.334756i0.000743602 + 0.334756i
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)
bad31 1 1
good2 1+(−0.5−0.363i)T+(0.618+1.90i)T2 1 + (-0.5 - 0.363i)T + (0.618 + 1.90i)T^{2}
3 1+(−1.12−0.502i)T+(2.00+2.22i)T2 1 + (-1.12 - 0.502i)T + (2.00 + 2.22i)T^{2}
5 1+(0.5−0.866i)T+(−2.5−4.33i)T2 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2}
7 1+(2.83−3.14i)T+(−0.731−6.96i)T2 1 + (2.83 - 3.14i)T + (-0.731 - 6.96i)T^{2}
11 1+(1.95+0.415i)T+(10.0+4.47i)T2 1 + (1.95 + 0.415i)T + (10.0 + 4.47i)T^{2}
13 1+(0.129+1.22i)T+(−12.7+2.70i)T2 1 + (0.129 + 1.22i)T + (-12.7 + 2.70i)T^{2}
17 1+(5.12−1.08i)T+(15.5−6.91i)T2 1 + (5.12 - 1.08i)T + (15.5 - 6.91i)T^{2}
19 1+(0.233−2.22i)T+(−18.5−3.95i)T2 1 + (0.233 - 2.22i)T + (-18.5 - 3.95i)T^{2}
23 1+(2.38−7.33i)T+(−18.6−13.5i)T2 1 + (2.38 - 7.33i)T + (-18.6 - 13.5i)T^{2}
29 1+(5.85+4.25i)T+(8.96+27.5i)T2 1 + (5.85 + 4.25i)T + (8.96 + 27.5i)T^{2}
37 1+(−1−1.73i)T+(−18.5+32.0i)T2 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2}
41 1+(−6.39+2.84i)T+(27.4−30.4i)T2 1 + (-6.39 + 2.84i)T + (27.4 - 30.4i)T^{2}
43 1+(−0.338+3.21i)T+(−42.0−8.94i)T2 1 + (-0.338 + 3.21i)T + (-42.0 - 8.94i)T^{2}
47 1+(−5.23+3.80i)T+(14.5−44.6i)T2 1 + (-5.23 + 3.80i)T + (14.5 - 44.6i)T^{2}
53 1+(1.02+1.13i)T+(−5.54+52.7i)T2 1 + (1.02 + 1.13i)T + (-5.54 + 52.7i)T^{2}
59 1+(2.04+0.909i)T+(39.4+43.8i)T2 1 + (2.04 + 0.909i)T + (39.4 + 43.8i)T^{2}
61 1+14.1T+61T2 1 + 14.1T + 61T^{2}
67 1+(4−6.92i)T+(−33.5−58.0i)T2 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2}
71 1+(−8.81−9.79i)T+(−7.42+70.6i)T2 1 + (-8.81 - 9.79i)T + (-7.42 + 70.6i)T^{2}
73 1+(−0.461−0.0981i)T+(66.6+29.6i)T2 1 + (-0.461 - 0.0981i)T + (66.6 + 29.6i)T^{2}
79 1+(1.67−0.355i)T+(72.1−32.1i)T2 1 + (1.67 - 0.355i)T + (72.1 - 32.1i)T^{2}
83 1+(−2.68+1.19i)T+(55.5−61.6i)T2 1 + (-2.68 + 1.19i)T + (55.5 - 61.6i)T^{2}
89 1+(0.527+1.62i)T+(−72.0+52.3i)T2 1 + (0.527 + 1.62i)T + (-72.0 + 52.3i)T^{2}
97 1+(−0.600−1.84i)T+(−78.4+57.0i)T2 1 + (-0.600 - 1.84i)T + (-78.4 + 57.0i)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

−10.26111021170137716719032812554, −9.310533564003306302283539267935, −9.168857453122780922389296356695, −8.004903380452685033283792271011, −6.88926599773031379321226716185, −5.87697502562249231931092221967, −5.59002823063009185678774977503, −4.08164493817898073026091465308, −3.24925498478259712805312822736, −2.21704725252552378320276771094, 0.11607423535526923119251637674, 2.33551682088031588461351314441, 3.10166109113870133955242627124, 4.20789034638780961464360002948, 4.75628134886213653314342625747, 6.35403752075445097354471516333, 7.28617286394662353249338386328, 7.87490988847403258260340992007, 8.792890192688992885925462769972, 9.351707584676160220520123063617

Graph of the ZZ-function along the critical line