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Properties

Label 2-31e2-31.16-c1-0-60
Degree 22
Conductor 961961
Sign −0.997−0.0753i-0.997 - 0.0753i
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  + (1.95 − 1.41i)2-s + (−1.95 − 1.41i)3-s + (1.18 − 3.64i)4-s + 5-s − 5.82·6-s + (0.746 − 2.29i)7-s + (−1.36 − 4.19i)8-s + (0.874 + 2.68i)9-s + (1.95 − 1.41i)10-s + (1.62 − 4.98i)11-s + (−7.47 + 5.43i)12-s + (1.47 + 1.07i)13-s + (−1.80 − 5.54i)14-s + (−1.95 − 1.41i)15-s + (−2.42 − 1.76i)16-s + (0.0530 + 0.163i)17-s + ⋯
L(s)  = 1  + (1.38 − 1.00i)2-s + (−1.12 − 0.819i)3-s + (0.591 − 1.82i)4-s + 0.447·5-s − 2.37·6-s + (0.281 − 0.867i)7-s + (−0.482 − 1.48i)8-s + (0.291 + 0.896i)9-s + (0.617 − 0.448i)10-s + (0.488 − 1.50i)11-s + (−2.15 + 1.56i)12-s + (0.410 + 0.298i)13-s + (−0.481 − 1.48i)14-s + (−0.504 − 0.366i)15-s + (−0.606 − 0.440i)16-s + (0.0128 + 0.0395i)17-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=((−0.997−0.0753i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0753i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=((−0.997−0.0753i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0753i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: −0.997−0.0753i-0.997 - 0.0753i
Analytic conductor: 7.673627.67362
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ961(388,⋅)\chi_{961} (388, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 961, ( :1/2), −0.997−0.0753i)(2,\ 961,\ (\ :1/2),\ -0.997 - 0.0753i)

Particular Values

L(1)L(1) ≈\approx 0.0965924+2.55851i0.0965924 + 2.55851i
L(12)L(\frac12) ≈\approx 0.0965924+2.55851i0.0965924 + 2.55851i
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+(−1.95+1.41i)T+(0.618−1.90i)T2 1 + (-1.95 + 1.41i)T + (0.618 - 1.90i)T^{2}
3 1+(1.95+1.41i)T+(0.927+2.85i)T2 1 + (1.95 + 1.41i)T + (0.927 + 2.85i)T^{2}
5 1−T+5T2 1 - T + 5T^{2}
7 1+(−0.746+2.29i)T+(−5.66−4.11i)T2 1 + (-0.746 + 2.29i)T + (-5.66 - 4.11i)T^{2}
11 1+(−1.62+4.98i)T+(−8.89−6.46i)T2 1 + (-1.62 + 4.98i)T + (-8.89 - 6.46i)T^{2}
13 1+(−1.47−1.07i)T+(4.01+12.3i)T2 1 + (-1.47 - 1.07i)T + (4.01 + 12.3i)T^{2}
17 1+(−0.0530−0.163i)T+(−13.7+9.99i)T2 1 + (-0.0530 - 0.163i)T + (-13.7 + 9.99i)T^{2}
19 1+(1.28−0.932i)T+(5.87−18.0i)T2 1 + (1.28 - 0.932i)T + (5.87 - 18.0i)T^{2}
23 1+(−1.23−3.80i)T+(−18.6+13.5i)T2 1 + (-1.23 - 3.80i)T + (-18.6 + 13.5i)T^{2}
29 1+(0.947−0.688i)T+(8.96−27.5i)T2 1 + (0.947 - 0.688i)T + (8.96 - 27.5i)T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1+(7.67−5.57i)T+(12.6−38.9i)T2 1 + (7.67 - 5.57i)T + (12.6 - 38.9i)T^{2}
43 1+(−7.19+5.23i)T+(13.2−40.8i)T2 1 + (-7.19 + 5.23i)T + (13.2 - 40.8i)T^{2}
47 1+(−1.34−0.973i)T+(14.5+44.6i)T2 1 + (-1.34 - 0.973i)T + (14.5 + 44.6i)T^{2}
53 1+(0.0530+0.163i)T+(−42.8+31.1i)T2 1 + (0.0530 + 0.163i)T + (-42.8 + 31.1i)T^{2}
59 1+(−8.14−5.91i)T+(18.2+56.1i)T2 1 + (-8.14 - 5.91i)T + (18.2 + 56.1i)T^{2}
61 1+2.82T+61T2 1 + 2.82T + 61T^{2}
67 1+5.24T+67T2 1 + 5.24T + 67T^{2}
71 1+(4.34+13.3i)T+(−57.4+41.7i)T2 1 + (4.34 + 13.3i)T + (-57.4 + 41.7i)T^{2}
73 1+(1.18−3.64i)T+(−59.0−42.9i)T2 1 + (1.18 - 3.64i)T + (-59.0 - 42.9i)T^{2}
79 1+(−4.71−14.4i)T+(−63.9+46.4i)T2 1 + (-4.71 - 14.4i)T + (-63.9 + 46.4i)T^{2}
83 1+(−3.29+2.39i)T+(25.6−78.9i)T2 1 + (-3.29 + 2.39i)T + (25.6 - 78.9i)T^{2}
89 1+(−3.85+11.8i)T+(−72.0−52.3i)T2 1 + (-3.85 + 11.8i)T + (-72.0 - 52.3i)T^{2}
97 1+(−3.34+10.2i)T+(−78.4−57.0i)T2 1 + (-3.34 + 10.2i)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

−10.20886148213416140093231755693, −8.909320678030390988994398726570, −7.59454033866082650085985133396, −6.50714548773836545000629915671, −5.93956030108375784518281806361, −5.31924054139595423858258647140, −4.14788155802545182707558147771, −3.32752993579582653685878773543, −1.76973640119156840354323532387, −0.926999746360359762691088287768, 2.27064705210076804052635103640, 3.83929580130220073589804698168, 4.64563993681762719192716358557, 5.22981707844418219675641441246, 5.95194659688360011055136543293, 6.57845386758434131250692681364, 7.53788282299152384627974133505, 8.719553289317090767563940607211, 9.734478116487142821882073120487, 10.51670613932175216559625908356

Graph of the ZZ-function along the critical line