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Properties

Label 2-31e2-31.16-c1-0-14
Degree 22
Conductor 961961
Sign −0.996+0.0889i-0.996 + 0.0889i
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  + (−0.309 + 0.224i)2-s + (1.85 + 1.34i)3-s + (−0.572 + 1.76i)4-s − 2.23·5-s − 0.874·6-s + (−0.309 + 0.951i)7-s + (−0.454 − 1.40i)8-s + (0.690 + 2.12i)9-s + (0.690 − 0.502i)10-s + (−1.31 + 4.03i)11-s + (−3.43 + 2.49i)12-s + (5.55 + 4.03i)13-s + (−0.118 − 0.363i)14-s + (−4.13 − 3.00i)15-s + (−2.54 − 1.84i)16-s + (−0.166 − 0.513i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.158i)2-s + (1.06 + 0.776i)3-s + (−0.286 + 0.881i)4-s − 0.999·5-s − 0.356·6-s + (−0.116 + 0.359i)7-s + (−0.160 − 0.495i)8-s + (0.230 + 0.708i)9-s + (0.218 − 0.158i)10-s + (−0.395 + 1.21i)11-s + (−0.990 + 0.719i)12-s + (1.54 + 1.11i)13-s + (−0.0315 − 0.0970i)14-s + (−1.06 − 0.776i)15-s + (−0.636 − 0.462i)16-s + (−0.0404 − 0.124i)17-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=((−0.996+0.0889i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0889i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=((−0.996+0.0889i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0889i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: −0.996+0.0889i-0.996 + 0.0889i
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), −0.996+0.0889i)(2,\ 961,\ (\ :1/2),\ -0.996 + 0.0889i)

Particular Values

L(1)L(1) ≈\approx 0.0504910−1.13330i0.0504910 - 1.13330i
L(12)L(\frac12) ≈\approx 0.0504910−1.13330i0.0504910 - 1.13330i
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+(0.309−0.224i)T+(0.618−1.90i)T2 1 + (0.309 - 0.224i)T + (0.618 - 1.90i)T^{2}
3 1+(−1.85−1.34i)T+(0.927+2.85i)T2 1 + (-1.85 - 1.34i)T + (0.927 + 2.85i)T^{2}
5 1+2.23T+5T2 1 + 2.23T + 5T^{2}
7 1+(0.309−0.951i)T+(−5.66−4.11i)T2 1 + (0.309 - 0.951i)T + (-5.66 - 4.11i)T^{2}
11 1+(1.31−4.03i)T+(−8.89−6.46i)T2 1 + (1.31 - 4.03i)T + (-8.89 - 6.46i)T^{2}
13 1+(−5.55−4.03i)T+(4.01+12.3i)T2 1 + (-5.55 - 4.03i)T + (4.01 + 12.3i)T^{2}
17 1+(0.166+0.513i)T+(−13.7+9.99i)T2 1 + (0.166 + 0.513i)T + (-13.7 + 9.99i)T^{2}
19 1+(0.809−0.587i)T+(5.87−18.0i)T2 1 + (0.809 - 0.587i)T + (5.87 - 18.0i)T^{2}
23 1+(2.12+6.52i)T+(−18.6+13.5i)T2 1 + (2.12 + 6.52i)T + (-18.6 + 13.5i)T^{2}
29 1+(2.99−2.17i)T+(8.96−27.5i)T2 1 + (2.99 - 2.17i)T + (8.96 - 27.5i)T^{2}
37 1+4.24T+37T2 1 + 4.24T + 37T^{2}
41 1+(6.04−4.39i)T+(12.6−38.9i)T2 1 + (6.04 - 4.39i)T + (12.6 - 38.9i)T^{2}
43 1+(0.166−0.121i)T+(13.2−40.8i)T2 1 + (0.166 - 0.121i)T + (13.2 - 40.8i)T^{2}
47 1+(3+2.17i)T+(14.5+44.6i)T2 1 + (3 + 2.17i)T + (14.5 + 44.6i)T^{2}
53 1+(1.62+4.98i)T+(−42.8+31.1i)T2 1 + (1.62 + 4.98i)T + (-42.8 + 31.1i)T^{2}
59 1+(−4.80−3.49i)T+(18.2+56.1i)T2 1 + (-4.80 - 3.49i)T + (18.2 + 56.1i)T^{2}
61 1+4.44T+61T2 1 + 4.44T + 61T^{2}
67 1−6T+67T2 1 - 6T + 67T^{2}
71 1+(−2.30−7.10i)T+(−57.4+41.7i)T2 1 + (-2.30 - 7.10i)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+(−3.43−10.5i)T+(−63.9+46.4i)T2 1 + (-3.43 - 10.5i)T + (-63.9 + 46.4i)T^{2}
83 1+(−2.55+1.85i)T+(25.6−78.9i)T2 1 + (-2.55 + 1.85i)T + (25.6 - 78.9i)T^{2}
89 1+(−1.81+5.57i)T+(−72.0−52.3i)T2 1 + (-1.81 + 5.57i)T + (-72.0 - 52.3i)T^{2}
97 1+(2.16−6.65i)T+(−78.4−57.0i)T2 1 + (2.16 - 6.65i)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

−10.16818983922799478845576521975, −9.385652689603866430350590357641, −8.547553691394672277361841467607, −8.363070450120405734665536416138, −7.33858353035506093180951143518, −6.49173487758495986595106999425, −4.71475160994951344202748073129, −4.01164100645533765330520596091, −3.48764152835142104301866320006, −2.26889770184985744718150736994, 0.50238138823720498035102120434, 1.70069504281993286228078324862, 3.22316670130601093682400870099, 3.78523006995041160247245904983, 5.38361719676496592721777031786, 6.16153184137514915843726361027, 7.36398449361435938711918190890, 8.210957972393553052281237864343, 8.451640120865079089093556652711, 9.405051388689390190814140926149

Graph of the ZZ-function along the critical line