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Properties

Label 2-728-13.10-c1-0-14
Degree 22
Conductor 728728
Sign 0.161+0.986i0.161 + 0.986i
Analytic cond. 5.813105.81310
Root an. cond. 2.411032.41103
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  + (0.0150 + 0.0261i)3-s − 0.691i·5-s + (−0.866 − 0.5i)7-s + (1.49 − 2.59i)9-s + (4.62 − 2.66i)11-s + (−3.55 + 0.629i)13-s + (0.0180 − 0.0104i)15-s + (−2.67 + 4.63i)17-s + (−3.86 − 2.23i)19-s − 0.0301i·21-s + (−2.95 − 5.12i)23-s + 4.52·25-s + 0.181·27-s + (−0.0499 − 0.0865i)29-s − 10.4i·31-s + ⋯
L(s)  = 1  + (0.00871 + 0.0150i)3-s − 0.309i·5-s + (−0.327 − 0.188i)7-s + (0.499 − 0.865i)9-s + (1.39 − 0.804i)11-s + (−0.984 + 0.174i)13-s + (0.00466 − 0.00269i)15-s + (−0.648 + 1.12i)17-s + (−0.887 − 0.512i)19-s − 0.00658i·21-s + (−0.617 − 1.06i)23-s + 0.904·25-s + 0.0348·27-s + (−0.00927 − 0.0160i)29-s − 1.87i·31-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=((0.161+0.986i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s+1/2) L(s)=((0.161+0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: 0.161+0.986i0.161 + 0.986i
Analytic conductor: 5.813105.81310
Root analytic conductor: 2.411032.41103
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ728(673,⋅)\chi_{728} (673, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 728, ( :1/2), 0.161+0.986i)(2,\ 728,\ (\ :1/2),\ 0.161 + 0.986i)

Particular Values

L(1)L(1) ≈\approx 1.03596−0.879888i1.03596 - 0.879888i
L(12)L(\frac12) ≈\approx 1.03596−0.879888i1.03596 - 0.879888i
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
7 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
13 1+(3.55−0.629i)T 1 + (3.55 - 0.629i)T
good3 1+(−0.0150−0.0261i)T+(−1.5+2.59i)T2 1 + (-0.0150 - 0.0261i)T + (-1.5 + 2.59i)T^{2}
5 1+0.691iT−5T2 1 + 0.691iT - 5T^{2}
11 1+(−4.62+2.66i)T+(5.5−9.52i)T2 1 + (-4.62 + 2.66i)T + (5.5 - 9.52i)T^{2}
17 1+(2.67−4.63i)T+(−8.5−14.7i)T2 1 + (2.67 - 4.63i)T + (-8.5 - 14.7i)T^{2}
19 1+(3.86+2.23i)T+(9.5+16.4i)T2 1 + (3.86 + 2.23i)T + (9.5 + 16.4i)T^{2}
23 1+(2.95+5.12i)T+(−11.5+19.9i)T2 1 + (2.95 + 5.12i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.0499+0.0865i)T+(−14.5+25.1i)T2 1 + (0.0499 + 0.0865i)T + (-14.5 + 25.1i)T^{2}
31 1+10.4iT−31T2 1 + 10.4iT - 31T^{2}
37 1+(−5.37+3.10i)T+(18.5−32.0i)T2 1 + (-5.37 + 3.10i)T + (18.5 - 32.0i)T^{2}
41 1+(−7.50+4.33i)T+(20.5−35.5i)T2 1 + (-7.50 + 4.33i)T + (20.5 - 35.5i)T^{2}
43 1+(−3.42+5.92i)T+(−21.5−37.2i)T2 1 + (-3.42 + 5.92i)T + (-21.5 - 37.2i)T^{2}
47 1−4.94iT−47T2 1 - 4.94iT - 47T^{2}
53 1+5.49T+53T2 1 + 5.49T + 53T^{2}
59 1+(−6.31−3.64i)T+(29.5+51.0i)T2 1 + (-6.31 - 3.64i)T + (29.5 + 51.0i)T^{2}
61 1+(−0.0818+0.141i)T+(−30.5−52.8i)T2 1 + (-0.0818 + 0.141i)T + (-30.5 - 52.8i)T^{2}
67 1+(11.2−6.48i)T+(33.5−58.0i)T2 1 + (11.2 - 6.48i)T + (33.5 - 58.0i)T^{2}
71 1+(4.35+2.51i)T+(35.5+61.4i)T2 1 + (4.35 + 2.51i)T + (35.5 + 61.4i)T^{2}
73 1−12.8iT−73T2 1 - 12.8iT - 73T^{2}
79 1−8.12T+79T2 1 - 8.12T + 79T^{2}
83 1+11.0iT−83T2 1 + 11.0iT - 83T^{2}
89 1+(−1.68+0.970i)T+(44.5−77.0i)T2 1 + (-1.68 + 0.970i)T + (44.5 - 77.0i)T^{2}
97 1+(−15.8−9.15i)T+(48.5+84.0i)T2 1 + (-15.8 - 9.15i)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

−10.15759066604450764125446000841, −9.138560459458672603070811391983, −8.824117956269984626200684488894, −7.53405528331494686749813918937, −6.47227671403744190223280503387, −6.08649159016698927613905232333, −4.34121868507540704387108958861, −3.96591643718808607247082036698, −2.36391025887752896744804329131, −0.72337146603765723112317076703, 1.71139241255683237059950520803, 2.88454786563821146866608576597, 4.29133648805382855880611513825, 4.99754119145837979041974050560, 6.39137424896062799870881894051, 7.05768195209904372374148876401, 7.84218737041494844414111578971, 9.091676624535626007923134468334, 9.692071756719286720343320815124, 10.49825532610852868091742621449

Graph of the ZZ-function along the critical line