↑

Properties

Label 2-588-7.4-c1-0-3
Degree 22
Conductor 588588
Sign 0.605+0.795i0.605 + 0.795i
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  + (−0.5 + 0.866i)3-s + (−1 − 1.73i)5-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 3·13-s + 1.99·15-s + (4 − 6.92i)17-s + (−0.5 − 0.866i)19-s + (−4 − 6.92i)23-s + (0.500 − 0.866i)25-s + 0.999·27-s + 4·29-s + (1.5 − 2.59i)31-s + (−0.999 − 1.73i)33-s + (0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.447 − 0.774i)5-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 0.832·13-s + 0.516·15-s + (0.970 − 1.68i)17-s + (−0.114 − 0.198i)19-s + (−0.834 − 1.44i)23-s + (0.100 − 0.173i)25-s + 0.192·27-s + 0.742·29-s + (0.269 − 0.466i)31-s + (−0.174 − 0.301i)33-s + (0.0821 + 0.142i)37-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((0.605+0.795i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+1/2) L(s)=((0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 0.605+0.795i0.605 + 0.795i
Analytic conductor: 4.695204.69520
Root analytic conductor: 2.166842.16684
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ588(361,⋅)\chi_{588} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :1/2), 0.605+0.795i)(2,\ 588,\ (\ :1/2),\ 0.605 + 0.795i)

Particular Values

L(1)L(1) ≈\approx 1.00696−0.499178i1.00696 - 0.499178i
L(12)L(\frac12) ≈\approx 1.00696−0.499178i1.00696 - 0.499178i
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
3 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
7 1 1
good5 1+(1+1.73i)T+(−2.5+4.33i)T2 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2}
11 1+(1−1.73i)T+(−5.5−9.52i)T2 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2}
13 1−3T+13T2 1 - 3T + 13T^{2}
17 1+(−4+6.92i)T+(−8.5−14.7i)T2 1 + (-4 + 6.92i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.5+0.866i)T+(−9.5+16.4i)T2 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2}
23 1+(4+6.92i)T+(−11.5+19.9i)T2 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2}
29 1−4T+29T2 1 - 4T + 29T^{2}
31 1+(−1.5+2.59i)T+(−15.5−26.8i)T2 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2}
37 1+(−0.5−0.866i)T+(−18.5+32.0i)T2 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−11T+43T2 1 - 11T + 43T^{2}
47 1+(−3−5.19i)T+(−23.5+40.7i)T2 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2}
53 1+(−6+10.3i)T+(−26.5−45.8i)T2 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2}
59 1+(−2+3.46i)T+(−29.5−51.0i)T2 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2}
61 1+(3+5.19i)T+(−30.5+52.8i)T2 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2}
67 1+(6.5−11.2i)T+(−33.5−58.0i)T2 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2}
71 1+10T+71T2 1 + 10T + 71T^{2}
73 1+(5.5−9.52i)T+(−36.5−63.2i)T2 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2}
79 1+(−1.5−2.59i)T+(−39.5+68.4i)T2 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2}
83 1+2T+83T2 1 + 2T + 83T^{2}
89 1+(−44.5+77.0i)T2 1 + (-44.5 + 77.0i)T^{2}
97 1+10T+97T2 1 + 10T + 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

−10.46581362455315820703270200404, −9.774325998618023450521222488917, −8.793829749877786594326541294749, −8.078314047019151047749279353773, −6.99022940396525788328374918806, −5.83681122472412503926110971329, −4.83450555496741839915063303491, −4.15554911104864310336645817503, −2.73572212186900775081370168887, −0.72500536789799164666846476553, 1.46196899898935132217137690139, 3.11440067935756924438277152789, 3.98277885183252021178572561827, 5.66878862298854949462179269936, 6.17381983040674805170905629738, 7.35687613609293183882966911755, 7.999192546505736906353541738593, 8.895836238776167463077657651039, 10.38655405434576491905132926398, 10.67474349052871886641334513973

Graph of the ZZ-function along the critical line