↑

Properties

Label 2-588-84.11-c1-0-15
Degree 22
Conductor 588588
Sign 0.752−0.658i0.752 - 0.658i
Analytic cond. 4.695204.69520
Root an. cond. 2.166842.16684
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  + (−1.34 − 0.443i)2-s + (−1.66 − 0.489i)3-s + (1.60 + 1.19i)4-s + (1.08 − 0.626i)5-s + (2.01 + 1.39i)6-s + (−1.63 − 2.31i)8-s + (2.52 + 1.62i)9-s + (−1.73 + 0.360i)10-s + (−2.52 + 4.38i)11-s + (−2.08 − 2.76i)12-s + 4.41·13-s + (−2.11 + 0.509i)15-s + (1.16 + 3.82i)16-s + (−5.06 − 2.92i)17-s + (−2.66 − 3.30i)18-s + (−1.32 + 0.765i)19-s + ⋯
L(s)  = 1  + (−0.949 − 0.313i)2-s + (−0.959 − 0.282i)3-s + (0.803 + 0.595i)4-s + (0.485 − 0.280i)5-s + (0.822 + 0.569i)6-s + (−0.576 − 0.817i)8-s + (0.840 + 0.542i)9-s + (−0.548 + 0.114i)10-s + (−0.762 + 1.32i)11-s + (−0.602 − 0.798i)12-s + 1.22·13-s + (−0.544 + 0.131i)15-s + (0.291 + 0.956i)16-s + (−1.22 − 0.709i)17-s + (−0.627 − 0.778i)18-s + (−0.304 + 0.175i)19-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((0.752−0.658i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+1/2) L(s)=((0.752−0.658i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 0.752−0.658i0.752 - 0.658i
Analytic conductor: 4.695204.69520
Root analytic conductor: 2.166842.16684
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ588(263,⋅)\chi_{588} (263, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :1/2), 0.752−0.658i)(2,\ 588,\ (\ :1/2),\ 0.752 - 0.658i)

Particular Values

L(1)L(1) ≈\approx 0.577831+0.217271i0.577831 + 0.217271i
L(12)L(\frac12) ≈\approx 0.577831+0.217271i0.577831 + 0.217271i
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+(1.34+0.443i)T 1 + (1.34 + 0.443i)T
3 1+(1.66+0.489i)T 1 + (1.66 + 0.489i)T
7 1 1
good5 1+(−1.08+0.626i)T+(2.5−4.33i)T2 1 + (-1.08 + 0.626i)T + (2.5 - 4.33i)T^{2}
11 1+(2.52−4.38i)T+(−5.5−9.52i)T2 1 + (2.52 - 4.38i)T + (-5.5 - 9.52i)T^{2}
13 1−4.41T+13T2 1 - 4.41T + 13T^{2}
17 1+(5.06+2.92i)T+(8.5+14.7i)T2 1 + (5.06 + 2.92i)T + (8.5 + 14.7i)T^{2}
19 1+(1.32−0.765i)T+(9.5−16.4i)T2 1 + (1.32 - 0.765i)T + (9.5 - 16.4i)T^{2}
23 1+(0.156+0.271i)T+(−11.5+19.9i)T2 1 + (0.156 + 0.271i)T + (-11.5 + 19.9i)T^{2}
29 1−5.53iT−29T2 1 - 5.53iT - 29T^{2}
31 1+(−4.24−2.44i)T+(15.5+26.8i)T2 1 + (-4.24 - 2.44i)T + (15.5 + 26.8i)T^{2}
37 1+(−2.66−4.62i)T+(−18.5+32.0i)T2 1 + (-2.66 - 4.62i)T + (-18.5 + 32.0i)T^{2}
41 1−1.97iT−41T2 1 - 1.97iT - 41T^{2}
43 1+3.18iT−43T2 1 + 3.18iT - 43T^{2}
47 1+(−5.74−9.95i)T+(−23.5+40.7i)T2 1 + (-5.74 - 9.95i)T + (-23.5 + 40.7i)T^{2}
53 1+(−11.0−6.39i)T+(26.5+45.8i)T2 1 + (-11.0 - 6.39i)T + (26.5 + 45.8i)T^{2}
59 1+(−3.98+6.89i)T+(−29.5−51.0i)T2 1 + (-3.98 + 6.89i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.151−0.262i)T+(−30.5+52.8i)T2 1 + (-0.151 - 0.262i)T + (-30.5 + 52.8i)T^{2}
67 1+(−9.13−5.27i)T+(33.5+58.0i)T2 1 + (-9.13 - 5.27i)T + (33.5 + 58.0i)T^{2}
71 1−4.53T+71T2 1 - 4.53T + 71T^{2}
73 1+(0.707−1.22i)T+(−36.5−63.2i)T2 1 + (0.707 - 1.22i)T + (-36.5 - 63.2i)T^{2}
79 1+(6−3.46i)T+(39.5−68.4i)T2 1 + (6 - 3.46i)T + (39.5 - 68.4i)T^{2}
83 1−6.78T+83T2 1 - 6.78T + 83T^{2}
89 1+(4.15−2.39i)T+(44.5−77.0i)T2 1 + (4.15 - 2.39i)T + (44.5 - 77.0i)T^{2}
97 1+13.6T+97T2 1 + 13.6T + 97T^{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

−10.76897480165550325475032716799, −10.02563866549123099806556877484, −9.200179836936141239940244521483, −8.202033646622765672710275827918, −7.16180684430210901781179502602, −6.52330656283968625315340475515, −5.41074119635240849362749981882, −4.26597777493307169461605636606, −2.42731642568118573213945742527, −1.28752800538067048306746358415, 0.58108509960016598243020219924, 2.29683693489468978217750463309, 3.99236208990847422644275291106, 5.57695782499185241554732666414, 6.08649022409611955971122361361, 6.77851356324497568732650911048, 8.147256853379764230917070393277, 8.782822861954873717959269274422, 9.888677174605853205038313952390, 10.65891217832609552826610723987

Graph of the ZZ-function along the critical line