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Properties

Label 2-31e2-31.16-c1-0-39
Degree 22
Conductor 961961
Sign 0.696−0.717i0.696 - 0.717i
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  + (0.809 − 0.587i)2-s + (2.28 + 1.66i)3-s + (−0.309 + 0.951i)4-s + 2.82·6-s + (1.23 − 3.80i)7-s + (0.927 + 2.85i)8-s + (1.54 + 4.75i)9-s + (−0.874 + 2.68i)11-s + (−2.28 + 1.66i)12-s + (1.14 + 0.831i)13-s + (−1.23 − 3.80i)14-s + (0.809 + 0.587i)16-s + (−0.437 − 1.34i)17-s + (4.04 + 2.93i)18-s + (3.23 − 2.35i)19-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (1.32 + 0.959i)3-s + (−0.154 + 0.475i)4-s + 1.15·6-s + (0.467 − 1.43i)7-s + (0.327 + 1.00i)8-s + (0.515 + 1.58i)9-s + (−0.263 + 0.811i)11-s + (−0.660 + 0.479i)12-s + (0.317 + 0.230i)13-s + (−0.330 − 1.01i)14-s + (0.202 + 0.146i)16-s + (−0.105 − 0.326i)17-s + (0.953 + 0.692i)18-s + (0.742 − 0.539i)19-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=((0.696−0.717i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=((0.696−0.717i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: 0.696−0.717i0.696 - 0.717i
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), 0.696−0.717i)(2,\ 961,\ (\ :1/2),\ 0.696 - 0.717i)

Particular Values

L(1)L(1) ≈\approx 3.06050+1.29554i3.06050 + 1.29554i
L(12)L(\frac12) ≈\approx 3.06050+1.29554i3.06050 + 1.29554i
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+(−0.809+0.587i)T+(0.618−1.90i)T2 1 + (-0.809 + 0.587i)T + (0.618 - 1.90i)T^{2}
3 1+(−2.28−1.66i)T+(0.927+2.85i)T2 1 + (-2.28 - 1.66i)T + (0.927 + 2.85i)T^{2}
5 1+5T2 1 + 5T^{2}
7 1+(−1.23+3.80i)T+(−5.66−4.11i)T2 1 + (-1.23 + 3.80i)T + (-5.66 - 4.11i)T^{2}
11 1+(0.874−2.68i)T+(−8.89−6.46i)T2 1 + (0.874 - 2.68i)T + (-8.89 - 6.46i)T^{2}
13 1+(−1.14−0.831i)T+(4.01+12.3i)T2 1 + (-1.14 - 0.831i)T + (4.01 + 12.3i)T^{2}
17 1+(0.437+1.34i)T+(−13.7+9.99i)T2 1 + (0.437 + 1.34i)T + (-13.7 + 9.99i)T^{2}
19 1+(−3.23+2.35i)T+(5.87−18.0i)T2 1 + (-3.23 + 2.35i)T + (5.87 - 18.0i)T^{2}
23 1+(−1.74−5.37i)T+(−18.6+13.5i)T2 1 + (-1.74 - 5.37i)T + (-18.6 + 13.5i)T^{2}
29 1+(1.14−0.831i)T+(8.96−27.5i)T2 1 + (1.14 - 0.831i)T + (8.96 - 27.5i)T^{2}
37 1−4.24T+37T2 1 - 4.24T + 37T^{2}
41 1+(−1.61+1.17i)T+(12.6−38.9i)T2 1 + (-1.61 + 1.17i)T + (12.6 - 38.9i)T^{2}
43 1+(−6.86+4.98i)T+(13.2−40.8i)T2 1 + (-6.86 + 4.98i)T + (13.2 - 40.8i)T^{2}
47 1+(9.70+7.05i)T+(14.5+44.6i)T2 1 + (9.70 + 7.05i)T + (14.5 + 44.6i)T^{2}
53 1+(1.31+4.03i)T+(−42.8+31.1i)T2 1 + (1.31 + 4.03i)T + (-42.8 + 31.1i)T^{2}
59 1+(6.47+4.70i)T+(18.2+56.1i)T2 1 + (6.47 + 4.70i)T + (18.2 + 56.1i)T^{2}
61 1+1.41T+61T2 1 + 1.41T + 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1+(−2.47−7.60i)T+(−57.4+41.7i)T2 1 + (-2.47 - 7.60i)T + (-57.4 + 41.7i)T^{2}
73 1+(−1.31+4.03i)T+(−59.0−42.9i)T2 1 + (-1.31 + 4.03i)T + (-59.0 - 42.9i)T^{2}
79 1+(−3.49−10.7i)T+(−63.9+46.4i)T2 1 + (-3.49 - 10.7i)T + (-63.9 + 46.4i)T^{2}
83 1+(11.4−8.31i)T+(25.6−78.9i)T2 1 + (11.4 - 8.31i)T + (25.6 - 78.9i)T^{2}
89 1+(−2.18+6.72i)T+(−72.0−52.3i)T2 1 + (-2.18 + 6.72i)T + (-72.0 - 52.3i)T^{2}
97 1+(−2.47+7.60i)T+(−78.4−57.0i)T2 1 + (-2.47 + 7.60i)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

−9.973538457741944584899075283971, −9.456542974500367003488807838364, −8.473555284978486689319950831527, −7.65404638109471015533330562658, −7.22689560205712922827832843839, −5.21346479651623890150686004070, −4.42438464802039533043708033997, −3.85189044362347456404444368818, −3.08903106456124521169085803083, −1.89654529688565128923031207327, 1.31113221548804069886282760625, 2.45206236445472566432441286515, 3.38895091248599708654057576514, 4.70459820410866436362668442383, 5.91493333197006132677696812781, 6.20650893802327818171339240025, 7.59356736467406730993444258708, 8.134009340486775372294123363965, 8.957808401426216296612891557754, 9.521362785936465617989696723439

Graph of the ZZ-function along the critical line