↑

Properties

Label 2-532-133.11-c1-0-7
Degree 22
Conductor 532532
Sign 0.0622−0.998i0.0622 - 0.998i
Analytic cond. 4.248044.24804
Root an. cond. 2.061072.06107
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.58 + 2.75i)3-s + 2.85·5-s + (1.33 + 2.28i)7-s + (−3.54 + 6.14i)9-s + (0.978 − 1.69i)11-s + (−2.81 − 4.86i)13-s + (4.52 + 7.84i)15-s + (0.0951 + 0.164i)17-s + (−1.31 − 4.15i)19-s + (−4.15 + 7.30i)21-s + (2.03 − 3.52i)23-s + 3.12·25-s − 13.0·27-s + (−3.37 − 5.84i)29-s + (2.76 − 4.78i)31-s + ⋯
L(s)  = 1  + (0.917 + 1.58i)3-s + 1.27·5-s + (0.505 + 0.862i)7-s + (−1.18 + 2.04i)9-s + (0.295 − 0.511i)11-s + (−0.779 − 1.35i)13-s + (1.16 + 2.02i)15-s + (0.0230 + 0.0399i)17-s + (−0.300 − 0.953i)19-s + (−0.906 + 1.59i)21-s + (0.423 − 0.734i)23-s + 0.624·25-s − 2.50·27-s + (−0.626 − 1.08i)29-s + (0.496 − 0.860i)31-s + ⋯

Functional equation

Λ(s)=(532s/2 ΓC(s) L(s)=((0.0622−0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0622 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(532s/2 ΓC(s+1/2) L(s)=((0.0622−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0622 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 532532    =    22⋅7⋅192^{2} \cdot 7 \cdot 19
Sign: 0.0622−0.998i0.0622 - 0.998i
Analytic conductor: 4.248044.24804
Root analytic conductor: 2.061072.06107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ532(277,⋅)\chi_{532} (277, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 532, ( :1/2), 0.0622−0.998i)(2,\ 532,\ (\ :1/2),\ 0.0622 - 0.998i)

Particular Values

L(1)L(1) ≈\approx 1.71011+1.60674i1.71011 + 1.60674i
L(12)L(\frac12) ≈\approx 1.71011+1.60674i1.71011 + 1.60674i
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)
bad2 1 1
7 1+(−1.33−2.28i)T 1 + (-1.33 - 2.28i)T
19 1+(1.31+4.15i)T 1 + (1.31 + 4.15i)T
good3 1+(−1.58−2.75i)T+(−1.5+2.59i)T2 1 + (-1.58 - 2.75i)T + (-1.5 + 2.59i)T^{2}
5 1−2.85T+5T2 1 - 2.85T + 5T^{2}
11 1+(−0.978+1.69i)T+(−5.5−9.52i)T2 1 + (-0.978 + 1.69i)T + (-5.5 - 9.52i)T^{2}
13 1+(2.81+4.86i)T+(−6.5+11.2i)T2 1 + (2.81 + 4.86i)T + (-6.5 + 11.2i)T^{2}
17 1+(−0.0951−0.164i)T+(−8.5+14.7i)T2 1 + (-0.0951 - 0.164i)T + (-8.5 + 14.7i)T^{2}
23 1+(−2.03+3.52i)T+(−11.5−19.9i)T2 1 + (-2.03 + 3.52i)T + (-11.5 - 19.9i)T^{2}
29 1+(3.37+5.84i)T+(−14.5+25.1i)T2 1 + (3.37 + 5.84i)T + (-14.5 + 25.1i)T^{2}
31 1+(−2.76+4.78i)T+(−15.5−26.8i)T2 1 + (-2.76 + 4.78i)T + (-15.5 - 26.8i)T^{2}
37 1+(1.08+1.87i)T+(−18.5+32.0i)T2 1 + (1.08 + 1.87i)T + (-18.5 + 32.0i)T^{2}
41 1+(5.78−10.0i)T+(−20.5−35.5i)T2 1 + (5.78 - 10.0i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.76−3.05i)T+(−21.5−37.2i)T2 1 + (1.76 - 3.05i)T + (-21.5 - 37.2i)T^{2}
47 1+(0.322−0.558i)T+(−23.5−40.7i)T2 1 + (0.322 - 0.558i)T + (-23.5 - 40.7i)T^{2}
53 1+0.986T+53T2 1 + 0.986T + 53T^{2}
59 1+(−0.517−0.896i)T+(−29.5+51.0i)T2 1 + (-0.517 - 0.896i)T + (-29.5 + 51.0i)T^{2}
61 1+(6.63−11.4i)T+(−30.5−52.8i)T2 1 + (6.63 - 11.4i)T + (-30.5 - 52.8i)T^{2}
67 1−2.98T+67T2 1 - 2.98T + 67T^{2}
71 1+(7.31−12.6i)T+(−35.5−61.4i)T2 1 + (7.31 - 12.6i)T + (-35.5 - 61.4i)T^{2}
73 1+(1.60+2.78i)T+(−36.5+63.2i)T2 1 + (1.60 + 2.78i)T + (-36.5 + 63.2i)T^{2}
79 1−17.3T+79T2 1 - 17.3T + 79T^{2}
83 1−1.00T+83T2 1 - 1.00T + 83T^{2}
89 1+(−3.33+5.77i)T+(−44.5−77.0i)T2 1 + (-3.33 + 5.77i)T + (-44.5 - 77.0i)T^{2}
97 1+(0.455−0.789i)T+(−48.5−84.0i)T2 1 + (0.455 - 0.789i)T + (-48.5 - 84.0i)T^{2}
show more
show less
   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

−10.72314756330852697939420789044, −9.965370673301687135988805875875, −9.403935993337369105546862174653, −8.659914257849886059524259452573, −7.903804305381126492364265594703, −6.08459246494268131365022120807, −5.28652644039624607935283778427, −4.52105786983106785535757846195, −2.98478209976894718050678025018, −2.35671507711614687214481510448, 1.61591283019238957079251268211, 1.95556623915889385782767018559, 3.54088342822400835745324628663, 5.10061552156946356607687691985, 6.47144636797934281639128793836, 6.99494733219979858426248760337, 7.75568618363911135200641007907, 8.870172072364436492261012996200, 9.497394581158860719693242673855, 10.50718795807322518410049657967

Graph of the ZZ-function along the critical line