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Properties

Label 2-592-37.9-c1-0-16
Degree 22
Conductor 592592
Sign −0.942+0.335i-0.942 + 0.335i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.453 − 2.56i)3-s + (0.356 + 0.299i)5-s + (−1.67 − 1.40i)7-s + (−3.57 − 1.30i)9-s + (−1.28 − 2.23i)11-s + (−1.81 + 0.659i)13-s + (0.930 − 0.780i)15-s + (−3.45 − 1.25i)17-s + (−0.195 + 1.11i)19-s + (−4.37 + 3.67i)21-s + (1.76 − 3.05i)23-s + (−0.830 − 4.71i)25-s + (−1.05 + 1.82i)27-s + (1.87 + 3.25i)29-s + 0.372·31-s + ⋯
L(s)  = 1  + (0.261 − 1.48i)3-s + (0.159 + 0.133i)5-s + (−0.634 − 0.532i)7-s + (−1.19 − 0.433i)9-s + (−0.388 − 0.673i)11-s + (−0.502 + 0.182i)13-s + (0.240 − 0.201i)15-s + (−0.837 − 0.304i)17-s + (−0.0449 + 0.254i)19-s + (−0.955 + 0.801i)21-s + (0.367 − 0.636i)23-s + (−0.166 − 0.942i)25-s + (−0.202 + 0.350i)27-s + (0.348 + 0.603i)29-s + 0.0668·31-s + ⋯

Functional equation

Λ(s)=(592s/2 ΓC(s) L(s)=((−0.942+0.335i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(592s/2 ΓC(s+1/2) L(s)=((−0.942+0.335i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 592592    =    24⋅372^{4} \cdot 37
Sign: −0.942+0.335i-0.942 + 0.335i
Analytic conductor: 4.727144.72714
Root analytic conductor: 2.174192.17419
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ592(305,⋅)\chi_{592} (305, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 592, ( :1/2), −0.942+0.335i)(2,\ 592,\ (\ :1/2),\ -0.942 + 0.335i)

Particular Values

L(1)L(1) ≈\approx 0.191070−1.10634i0.191070 - 1.10634i
L(12)L(\frac12) ≈\approx 0.191070−1.10634i0.191070 - 1.10634i
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+(−5.93−1.31i)T 1 + (-5.93 - 1.31i)T
good3 1+(−0.453+2.56i)T+(−2.81−1.02i)T2 1 + (-0.453 + 2.56i)T + (-2.81 - 1.02i)T^{2}
5 1+(−0.356−0.299i)T+(0.868+4.92i)T2 1 + (-0.356 - 0.299i)T + (0.868 + 4.92i)T^{2}
7 1+(1.67+1.40i)T+(1.21+6.89i)T2 1 + (1.67 + 1.40i)T + (1.21 + 6.89i)T^{2}
11 1+(1.28+2.23i)T+(−5.5+9.52i)T2 1 + (1.28 + 2.23i)T + (-5.5 + 9.52i)T^{2}
13 1+(1.81−0.659i)T+(9.95−8.35i)T2 1 + (1.81 - 0.659i)T + (9.95 - 8.35i)T^{2}
17 1+(3.45+1.25i)T+(13.0+10.9i)T2 1 + (3.45 + 1.25i)T + (13.0 + 10.9i)T^{2}
19 1+(0.195−1.11i)T+(−17.8−6.49i)T2 1 + (0.195 - 1.11i)T + (-17.8 - 6.49i)T^{2}
23 1+(−1.76+3.05i)T+(−11.5−19.9i)T2 1 + (-1.76 + 3.05i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.87−3.25i)T+(−14.5+25.1i)T2 1 + (-1.87 - 3.25i)T + (-14.5 + 25.1i)T^{2}
31 1−0.372T+31T2 1 - 0.372T + 31T^{2}
41 1+(−7.81+2.84i)T+(31.4−26.3i)T2 1 + (-7.81 + 2.84i)T + (31.4 - 26.3i)T^{2}
43 1+9.43T+43T2 1 + 9.43T + 43T^{2}
47 1+(2.11−3.67i)T+(−23.5−40.7i)T2 1 + (2.11 - 3.67i)T + (-23.5 - 40.7i)T^{2}
53 1+(7.09−5.95i)T+(9.20−52.1i)T2 1 + (7.09 - 5.95i)T + (9.20 - 52.1i)T^{2}
59 1+(3.98−3.34i)T+(10.2−58.1i)T2 1 + (3.98 - 3.34i)T + (10.2 - 58.1i)T^{2}
61 1+(−12.3+4.50i)T+(46.7−39.2i)T2 1 + (-12.3 + 4.50i)T + (46.7 - 39.2i)T^{2}
67 1+(10.4+8.73i)T+(11.6+65.9i)T2 1 + (10.4 + 8.73i)T + (11.6 + 65.9i)T^{2}
71 1+(−2.05+11.6i)T+(−66.7−24.2i)T2 1 + (-2.05 + 11.6i)T + (-66.7 - 24.2i)T^{2}
73 1−1.32T+73T2 1 - 1.32T + 73T^{2}
79 1+(−13.0−10.9i)T+(13.7+77.7i)T2 1 + (-13.0 - 10.9i)T + (13.7 + 77.7i)T^{2}
83 1+(−12.4−4.54i)T+(63.5+53.3i)T2 1 + (-12.4 - 4.54i)T + (63.5 + 53.3i)T^{2}
89 1+(−11.7+9.84i)T+(15.4−87.6i)T2 1 + (-11.7 + 9.84i)T + (15.4 - 87.6i)T^{2}
97 1+(−4.27+7.40i)T+(−48.5−84.0i)T2 1 + (-4.27 + 7.40i)T + (-48.5 - 84.0i)T^{2}
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   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

−10.36288176796275893039204828904, −9.302126719462339530625796985692, −8.312750407823377605720067743479, −7.57305430131222601742646897993, −6.64762759451849475899699936039, −6.20616693372570800749714221475, −4.68857425915164171179141156285, −3.14585684154843485433817641387, −2.17528283964442668346607311065, −0.58137469277643524338074613814, 2.40405906049935035495795566968, 3.47600432272654766371711895347, 4.57614427621028716051537755123, 5.28875971563942790479892310802, 6.42251710698270209764854046978, 7.67141534915541385001076352226, 8.802416533014729090542578825033, 9.524286986183979162906173265150, 9.930380207428329722109899214322, 10.88487495025554616299943752696

Graph of the ZZ-function along the critical line