↑

Properties

Label 2-588-21.20-c5-0-15
Degree 22
Conductor 588588
Sign −0.488+0.872i-0.488 + 0.872i
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  + (−1.39 + 15.5i)3-s − 69.8·5-s + (−239. − 43.2i)9-s + 465. i·11-s + 285. i·13-s + (97.2 − 1.08e3i)15-s + 898.·17-s + 2.09e3i·19-s + 4.90e3i·23-s + 1.76e3·25-s + (1.00e3 − 3.65e3i)27-s + 5.87e3i·29-s + 3.85e3i·31-s + (−7.23e3 − 648. i)33-s − 3.61e3·37-s + ⋯
L(s)  = 1  + (−0.0892 + 0.996i)3-s − 1.25·5-s + (−0.984 − 0.177i)9-s + 1.16i·11-s + 0.469i·13-s + (0.111 − 1.24i)15-s + 0.753·17-s + 1.33i·19-s + 1.93i·23-s + 0.563·25-s + (0.264 − 0.964i)27-s + 1.29i·29-s + 0.720i·31-s + (−1.15 − 0.103i)33-s − 0.434·37-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((−0.488+0.872i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+5/2) L(s)=((−0.488+0.872i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: −0.488+0.872i-0.488 + 0.872i
Analytic conductor: 94.305694.3056
Root analytic conductor: 9.711119.71111
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ588(293,⋅)\chi_{588} (293, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :5/2), −0.488+0.872i)(2,\ 588,\ (\ :5/2),\ -0.488 + 0.872i)

Particular Values

L(3)L(3) ≈\approx 0.89099224960.8909922496
L(12)L(\frac12) ≈\approx 0.89099224960.8909922496
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+(1.39−15.5i)T 1 + (1.39 - 15.5i)T
7 1 1
good5 1+69.8T+3.12e3T2 1 + 69.8T + 3.12e3T^{2}
11 1−465.iT−1.61e5T2 1 - 465. iT - 1.61e5T^{2}
13 1−285.iT−3.71e5T2 1 - 285. iT - 3.71e5T^{2}
17 1−898.T+1.41e6T2 1 - 898.T + 1.41e6T^{2}
19 1−2.09e3iT−2.47e6T2 1 - 2.09e3iT - 2.47e6T^{2}
23 1−4.90e3iT−6.43e6T2 1 - 4.90e3iT - 6.43e6T^{2}
29 1−5.87e3iT−2.05e7T2 1 - 5.87e3iT - 2.05e7T^{2}
31 1−3.85e3iT−2.86e7T2 1 - 3.85e3iT - 2.86e7T^{2}
37 1+3.61e3T+6.93e7T2 1 + 3.61e3T + 6.93e7T^{2}
41 1−1.71e4T+1.15e8T2 1 - 1.71e4T + 1.15e8T^{2}
43 1+1.85e4T+1.47e8T2 1 + 1.85e4T + 1.47e8T^{2}
47 1−2.42e3T+2.29e8T2 1 - 2.42e3T + 2.29e8T^{2}
53 1−7.61e3iT−4.18e8T2 1 - 7.61e3iT - 4.18e8T^{2}
59 1+1.06e4T+7.14e8T2 1 + 1.06e4T + 7.14e8T^{2}
61 1−1.56e4iT−8.44e8T2 1 - 1.56e4iT - 8.44e8T^{2}
67 1+2.97e3T+1.35e9T2 1 + 2.97e3T + 1.35e9T^{2}
71 1−1.62e4iT−1.80e9T2 1 - 1.62e4iT - 1.80e9T^{2}
73 1−1.46e4iT−2.07e9T2 1 - 1.46e4iT - 2.07e9T^{2}
79 1+8.61e4T+3.07e9T2 1 + 8.61e4T + 3.07e9T^{2}
83 1−2.74e3T+3.93e9T2 1 - 2.74e3T + 3.93e9T^{2}
89 1−1.01e5T+5.58e9T2 1 - 1.01e5T + 5.58e9T^{2}
97 1+2.34e4iT−8.58e9T2 1 + 2.34e4iT - 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

−10.39512350896279161197536456910, −9.733150763832826412061566985746, −8.828478928837068092911129909519, −7.82544645233373747552826923453, −7.16569626793204863244131868543, −5.73802115678882106231296564021, −4.80765127272082477422119370976, −3.89265128049339003387854445969, −3.26832918691912257069371712532, −1.51040807852404602037953738804, 0.31006399939765151973950312930, 0.70588602950228795784702917118, 2.48874162263175791447334564673, 3.39241383492159490719642501602, 4.62257019196225831524784769352, 5.85136091702490945336723380382, 6.68355823184572880381317824421, 7.72396213987259566361029267738, 8.181350894227954265885153620129, 8.992134806611633503920996494205

Graph of the ZZ-function along the critical line