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Properties

Label 2-31e2-31.7-c1-0-13
Degree 22
Conductor 961961
Sign −0.976+0.213i-0.976 + 0.213i
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 + 1.53i)2-s + (−3.16 + 0.672i)3-s + (−0.5 + 0.363i)4-s + (−0.5 + 0.866i)5-s + (−2.61 − 4.53i)6-s + (0.215 + 0.0960i)7-s + (1.80 + 1.31i)8-s + (6.82 − 3.03i)9-s + (−1.58 − 0.336i)10-s + (0.209 + 1.98i)11-s + (1.33 − 1.48i)12-s + (2.16 + 2.40i)13-s + (−0.0399 + 0.379i)14-s + (1 − 3.07i)15-s + (−1.50 + 4.61i)16-s + (0.0798 − 0.759i)17-s + ⋯
L(s)  = 1  + (0.353 + 1.08i)2-s + (−1.82 + 0.388i)3-s + (−0.250 + 0.181i)4-s + (−0.223 + 0.387i)5-s + (−1.06 − 1.85i)6-s + (0.0815 + 0.0362i)7-s + (0.639 + 0.464i)8-s + (2.27 − 1.01i)9-s + (−0.500 − 0.106i)10-s + (0.0630 + 0.599i)11-s + (0.386 − 0.429i)12-s + (0.600 + 0.666i)13-s + (−0.0106 + 0.101i)14-s + (0.258 − 0.794i)15-s + (−0.375 + 1.15i)16-s + (0.0193 − 0.184i)17-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=((−0.976+0.213i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=((−0.976+0.213i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: −0.976+0.213i-0.976 + 0.213i
Analytic conductor: 7.673627.67362
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ961(844,⋅)\chi_{961} (844, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 961, ( :1/2), −0.976+0.213i)(2,\ 961,\ (\ :1/2),\ -0.976 + 0.213i)

Particular Values

L(1)L(1) ≈\approx 0.105237−0.973759i0.105237 - 0.973759i
L(12)L(\frac12) ≈\approx 0.105237−0.973759i0.105237 - 0.973759i
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+(−0.5−1.53i)T+(−1.61+1.17i)T2 1 + (-0.5 - 1.53i)T + (-1.61 + 1.17i)T^{2}
3 1+(3.16−0.672i)T+(2.74−1.22i)T2 1 + (3.16 - 0.672i)T + (2.74 - 1.22i)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+(−0.215−0.0960i)T+(4.68+5.20i)T2 1 + (-0.215 - 0.0960i)T + (4.68 + 5.20i)T^{2}
11 1+(−0.209−1.98i)T+(−10.7+2.28i)T2 1 + (-0.209 - 1.98i)T + (-10.7 + 2.28i)T^{2}
13 1+(−2.16−2.40i)T+(−1.35+12.9i)T2 1 + (-2.16 - 2.40i)T + (-1.35 + 12.9i)T^{2}
17 1+(−0.0798+0.759i)T+(−16.6−3.53i)T2 1 + (-0.0798 + 0.759i)T + (-16.6 - 3.53i)T^{2}
19 1+(1.49−1.66i)T+(−1.98−18.8i)T2 1 + (1.49 - 1.66i)T + (-1.98 - 18.8i)T^{2}
23 1+(−4.61−3.35i)T+(7.10+21.8i)T2 1 + (-4.61 - 3.35i)T + (7.10 + 21.8i)T^{2}
29 1+(0.854+2.62i)T+(−23.4+17.0i)T2 1 + (0.854 + 2.62i)T + (-23.4 + 17.0i)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+(6.84+1.45i)T+(37.4+16.6i)T2 1 + (6.84 + 1.45i)T + (37.4 + 16.6i)T^{2}
43 1+(0.827−0.918i)T+(−4.49−42.7i)T2 1 + (0.827 - 0.918i)T + (-4.49 - 42.7i)T^{2}
47 1+(−0.763+2.35i)T+(−38.0−27.6i)T2 1 + (-0.763 + 2.35i)T + (-38.0 - 27.6i)T^{2}
53 1+(−9.56+4.25i)T+(35.4−39.3i)T2 1 + (-9.56 + 4.25i)T + (35.4 - 39.3i)T^{2}
59 1+(2.18−0.464i)T+(53.8−23.9i)T2 1 + (2.18 - 0.464i)T + (53.8 - 23.9i)T^{2}
61 1+8.18T+61T2 1 + 8.18T + 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+(8.38−3.73i)T+(47.5−52.7i)T2 1 + (8.38 - 3.73i)T + (47.5 - 52.7i)T^{2}
73 1+(−0.885−8.42i)T+(−71.4+15.1i)T2 1 + (-0.885 - 8.42i)T + (-71.4 + 15.1i)T^{2}
79 1+(1.22−11.6i)T+(−77.2−16.4i)T2 1 + (1.22 - 11.6i)T + (-77.2 - 16.4i)T^{2}
83 1+(14.6+3.10i)T+(75.8+33.7i)T2 1 + (14.6 + 3.10i)T + (75.8 + 33.7i)T^{2}
89 1+(−9.47+6.88i)T+(27.5−84.6i)T2 1 + (-9.47 + 6.88i)T + (27.5 - 84.6i)T^{2}
97 1+(−12.8+9.37i)T+(29.9−92.2i)T2 1 + (-12.8 + 9.37i)T + (29.9 - 92.2i)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

−10.59215839523724529612017315659, −9.891661296549121398032222949144, −8.710563897516164088892071144164, −7.27516145714776935385383764415, −6.97874620203888313906928629832, −6.14200069447733079832408734410, −5.42326047624162397477317526747, −4.72042028204797129317005884926, −3.83449799193111635304736941703, −1.54690774430578196714892649507, 0.56906579653739646137767397002, 1.50705092370514977211226580736, 3.10556657297953477120340437343, 4.39448573548930614357429059339, 4.99409119905619728978412472071, 6.04593272225681182463781859014, 6.78329304887953857193893509408, 7.73811689747419667447122409016, 8.874918039595230097936793655513, 10.27136277838794688899868199713

Graph of the ZZ-function along the critical line