↑

Properties

Label 2-588-7.2-c5-0-19
Degree 22
Conductor 588588
Sign −0.266+0.963i-0.266 + 0.963i
Analytic cond. 94.305694.3056
Root an. cond. 9.711119.71111
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 − 7.79i)3-s + (−38.8 + 67.2i)5-s + (−40.5 + 70.1i)9-s + (−238. − 413. i)11-s + 63.7·13-s + 698.·15-s + (518. + 898. i)17-s + (−333. + 577. i)19-s + (−1.62e3 + 2.81e3i)23-s + (−1.45e3 − 2.51e3i)25-s + 729·27-s + 2.30e3·29-s + (1.85e3 + 3.21e3i)31-s + (−2.14e3 + 3.72e3i)33-s + (−6.12e3 + 1.06e4i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.694 + 1.20i)5-s + (−0.166 + 0.288i)9-s + (−0.594 − 1.03i)11-s + 0.104·13-s + 0.801·15-s + (0.435 + 0.754i)17-s + (−0.211 + 0.367i)19-s + (−0.640 + 1.10i)23-s + (−0.464 − 0.803i)25-s + 0.192·27-s + 0.508·29-s + (0.347 + 0.601i)31-s + (−0.343 + 0.594i)33-s + (−0.735 + 1.27i)37-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((−0.266+0.963i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+5/2) L(s)=((−0.266+0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: −0.266+0.963i-0.266 + 0.963i
Analytic conductor: 94.305694.3056
Root analytic conductor: 9.711119.71111
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ588(373,⋅)\chi_{588} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :5/2), −0.266+0.963i)(2,\ 588,\ (\ :5/2),\ -0.266 + 0.963i)

Particular Values

L(3)L(3) ≈\approx 0.38565947920.3856594792
L(12)L(\frac12) ≈\approx 0.38565947920.3856594792
L(72)L(\frac{7}{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+(4.5+7.79i)T 1 + (4.5 + 7.79i)T
7 1 1
good5 1+(38.8−67.2i)T+(−1.56e3−2.70e3i)T2 1 + (38.8 - 67.2i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(238.+413.i)T+(−8.05e4+1.39e5i)T2 1 + (238. + 413. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−63.7T+3.71e5T2 1 - 63.7T + 3.71e5T^{2}
17 1+(−518.−898.i)T+(−7.09e5+1.22e6i)T2 1 + (-518. - 898. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(333.−577.i)T+(−1.23e6−2.14e6i)T2 1 + (333. - 577. i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(1.62e3−2.81e3i)T+(−3.21e6−5.57e6i)T2 1 + (1.62e3 - 2.81e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1−2.30e3T+2.05e7T2 1 - 2.30e3T + 2.05e7T^{2}
31 1+(−1.85e3−3.21e3i)T+(−1.43e7+2.47e7i)T2 1 + (-1.85e3 - 3.21e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(6.12e3−1.06e4i)T+(−3.46e7−6.00e7i)T2 1 + (6.12e3 - 1.06e4i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1−1.82e3T+1.15e8T2 1 - 1.82e3T + 1.15e8T^{2}
43 1+2.07e4T+1.47e8T2 1 + 2.07e4T + 1.47e8T^{2}
47 1+(2.14e3−3.70e3i)T+(−1.14e8−1.98e8i)T2 1 + (2.14e3 - 3.70e3i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(1.28e4+2.22e4i)T+(−2.09e8+3.62e8i)T2 1 + (1.28e4 + 2.22e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(1.41e3+2.45e3i)T+(−3.57e8+6.19e8i)T2 1 + (1.41e3 + 2.45e3i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−8.40e3+1.45e4i)T+(−4.22e8−7.31e8i)T2 1 + (-8.40e3 + 1.45e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(−3.12e4−5.41e4i)T+(−6.75e8+1.16e9i)T2 1 + (-3.12e4 - 5.41e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1−7.23e4T+1.80e9T2 1 - 7.23e4T + 1.80e9T^{2}
73 1+(2.78e4+4.82e4i)T+(−1.03e9+1.79e9i)T2 1 + (2.78e4 + 4.82e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(−1.99e3+3.45e3i)T+(−1.53e9−2.66e9i)T2 1 + (-1.99e3 + 3.45e3i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−4.60e4T+3.93e9T2 1 - 4.60e4T + 3.93e9T^{2}
89 1+(−6.76e4+1.17e5i)T+(−2.79e9−4.83e9i)T2 1 + (-6.76e4 + 1.17e5i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+1.42e5T+8.58e9T2 1 + 1.42e5T + 8.58e9T^{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

−9.969679165019472652208172332690, −8.340107497236236323854275732585, −7.975738278175087615507194311913, −6.88635258917440491606022157397, −6.21697453981815996563972831298, −5.19916182954355879991555913375, −3.64750509943567458252656660484, −3.02498987225598824498461885088, −1.59804112156089165665430964074, −0.11862014060307787771893686478, 0.799937453075725890137391300786, 2.33762802900758455706464986772, 3.82673394934010298392069978200, 4.71910719493133869426373524742, 5.21062515158322478743735948884, 6.56699996784460935574517593423, 7.69211452236991929118294298481, 8.426269252361693004690008995373, 9.330418730424216858356807112519, 10.10040851420790273398255029053

Graph of the ZZ-function along the critical line