↑

Properties

Label 2-592-148.31-c1-0-5
Degree 22
Conductor 592592
Sign 0.763−0.646i0.763 - 0.646i
Analytic cond. 4.727144.72714
Root an. cond. 2.174192.17419
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  − 2.64·3-s + (−1 + i)5-s − 2.64i·7-s + 4.00·9-s − 2.64·11-s + (−1 + i)13-s + (2.64 − 2.64i)15-s + (2 − 2i)17-s + 7.00i·21-s + (2.64 + 2.64i)23-s + 3i·25-s − 2.64·27-s + (7 + 7i)29-s + 7.00·33-s + (2.64 + 2.64i)35-s + ⋯
L(s)  = 1  − 1.52·3-s + (−0.447 + 0.447i)5-s − 0.999i·7-s + 1.33·9-s − 0.797·11-s + (−0.277 + 0.277i)13-s + (0.683 − 0.683i)15-s + (0.485 − 0.485i)17-s + 1.52i·21-s + (0.551 + 0.551i)23-s + 0.600i·25-s − 0.509·27-s + (1.29 + 1.29i)29-s + 1.21·33-s + (0.447 + 0.447i)35-s + ⋯

Functional equation

Λ(s)=(592s/2 ΓC(s) L(s)=((0.763−0.646i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(592s/2 ΓC(s+1/2) L(s)=((0.763−0.646i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 592592    =    24⋅372^{4} \cdot 37
Sign: 0.763−0.646i0.763 - 0.646i
Analytic conductor: 4.727144.72714
Root analytic conductor: 2.174192.17419
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ592(31,⋅)\chi_{592} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 592, ( :1/2), 0.763−0.646i)(2,\ 592,\ (\ :1/2),\ 0.763 - 0.646i)

Particular Values

L(1)L(1) ≈\approx 0.600498+0.220160i0.600498 + 0.220160i
L(12)L(\frac12) ≈\approx 0.600498+0.220160i0.600498 + 0.220160i
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
37 1+(−1+6i)T 1 + (-1 + 6i)T
good3 1+2.64T+3T2 1 + 2.64T + 3T^{2}
5 1+(1−i)T−5iT2 1 + (1 - i)T - 5iT^{2}
7 1+2.64iT−7T2 1 + 2.64iT - 7T^{2}
11 1+2.64T+11T2 1 + 2.64T + 11T^{2}
13 1+(1−i)T−13iT2 1 + (1 - i)T - 13iT^{2}
17 1+(−2+2i)T−17iT2 1 + (-2 + 2i)T - 17iT^{2}
19 1+19iT2 1 + 19iT^{2}
23 1+(−2.64−2.64i)T+23iT2 1 + (-2.64 - 2.64i)T + 23iT^{2}
29 1+(−7−7i)T+29iT2 1 + (-7 - 7i)T + 29iT^{2}
31 1−31iT2 1 - 31iT^{2}
41 1−3iT−41T2 1 - 3iT - 41T^{2}
43 1+(−7.93−7.93i)T+43iT2 1 + (-7.93 - 7.93i)T + 43iT^{2}
47 1−7.93iT−47T2 1 - 7.93iT - 47T^{2}
53 1−3T+53T2 1 - 3T + 53T^{2}
59 1+(−5.29−5.29i)T+59iT2 1 + (-5.29 - 5.29i)T + 59iT^{2}
61 1+(6+6i)T+61iT2 1 + (6 + 6i)T + 61iT^{2}
67 1+5.29T+67T2 1 + 5.29T + 67T^{2}
71 1−2.64iT−71T2 1 - 2.64iT - 71T^{2}
73 1+15iT−73T2 1 + 15iT - 73T^{2}
79 1+(−5.29−5.29i)T+79iT2 1 + (-5.29 - 5.29i)T + 79iT^{2}
83 1+13.2iT−83T2 1 + 13.2iT - 83T^{2}
89 1+(−4−4i)T+89iT2 1 + (-4 - 4i)T + 89iT^{2}
97 1+(−6+6i)T−97iT2 1 + (-6 + 6i)T - 97iT^{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

−10.79647454982643989393612904684, −10.36735765566232031661927404125, −9.257896559051280656395406034196, −7.65744431100030758845054161817, −7.22408801751207204610309097889, −6.26169152924263748092345048178, −5.21615931314666885300694326619, −4.45823645112189294637637164889, −3.11597850888458860760440926805, −0.959187043391705670538466054835, 0.59510735106584512424673630890, 2.56104864384797019233987783835, 4.31131575419316987530905883747, 5.22275929305860832064320969445, 5.81569157495462038847668696402, 6.76774666521787070377219255644, 7.979843858296724315537815128746, 8.726665904173579082644473728943, 10.06494778022201382938784444349, 10.59305676990730435575837863332

Graph of the ZZ-function along the critical line