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Properties

Label 2-735-105.32-c1-0-61
Degree 22
Conductor 735735
Sign 0.996+0.0821i0.996 + 0.0821i
Analytic cond. 5.869005.86900
Root an. cond. 2.422602.42260
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  + (2.45 + 0.658i)2-s + (1.25 − 1.19i)3-s + (3.87 + 2.23i)4-s + (−1.98 − 1.02i)5-s + (3.87 − 2.10i)6-s + (4.45 + 4.45i)8-s + (0.154 − 2.99i)9-s + (−4.21 − 3.82i)10-s + (1.35 + 0.784i)11-s + (7.53 − 1.81i)12-s + (2.21 − 2.21i)13-s + (−3.71 + 1.08i)15-s + (3.54 + 6.14i)16-s + (1.32 + 4.92i)17-s + (2.35 − 7.26i)18-s + (−1.45 + 0.840i)19-s + ⋯
L(s)  = 1  + (1.73 + 0.465i)2-s + (0.725 − 0.688i)3-s + (1.93 + 1.11i)4-s + (−0.889 − 0.457i)5-s + (1.58 − 0.859i)6-s + (1.57 + 1.57i)8-s + (0.0514 − 0.998i)9-s + (−1.33 − 1.20i)10-s + (0.409 + 0.236i)11-s + (2.17 − 0.523i)12-s + (0.615 − 0.615i)13-s + (−0.959 + 0.280i)15-s + (0.886 + 1.53i)16-s + (0.320 + 1.19i)17-s + (0.554 − 1.71i)18-s + (−0.333 + 0.192i)19-s + ⋯

Functional equation

Λ(s)=(735s/2 ΓC(s) L(s)=((0.996+0.0821i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0821i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(735s/2 ΓC(s+1/2) L(s)=((0.996+0.0821i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0821i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 735735    =    3⋅5⋅723 \cdot 5 \cdot 7^{2}
Sign: 0.996+0.0821i0.996 + 0.0821i
Analytic conductor: 5.869005.86900
Root analytic conductor: 2.422602.42260
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ735(557,⋅)\chi_{735} (557, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 735, ( :1/2), 0.996+0.0821i)(2,\ 735,\ (\ :1/2),\ 0.996 + 0.0821i)

Particular Values

L(1)L(1) ≈\approx 4.58546−0.188732i4.58546 - 0.188732i
L(12)L(\frac12) ≈\approx 4.58546−0.188732i4.58546 - 0.188732i
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)
bad3 1+(−1.25+1.19i)T 1 + (-1.25 + 1.19i)T
5 1+(1.98+1.02i)T 1 + (1.98 + 1.02i)T
7 1 1
good2 1+(−2.45−0.658i)T+(1.73+i)T2 1 + (-2.45 - 0.658i)T + (1.73 + i)T^{2}
11 1+(−1.35−0.784i)T+(5.5+9.52i)T2 1 + (-1.35 - 0.784i)T + (5.5 + 9.52i)T^{2}
13 1+(−2.21+2.21i)T−13iT2 1 + (-2.21 + 2.21i)T - 13iT^{2}
17 1+(−1.32−4.92i)T+(−14.7+8.5i)T2 1 + (-1.32 - 4.92i)T + (-14.7 + 8.5i)T^{2}
19 1+(1.45−0.840i)T+(9.5−16.4i)T2 1 + (1.45 - 0.840i)T + (9.5 - 16.4i)T^{2}
23 1+(−0.364+1.36i)T+(−19.9−11.5i)T2 1 + (-0.364 + 1.36i)T + (-19.9 - 11.5i)T^{2}
29 1+8.91T+29T2 1 + 8.91T + 29T^{2}
31 1+(1.37−2.38i)T+(−15.5−26.8i)T2 1 + (1.37 - 2.38i)T + (-15.5 - 26.8i)T^{2}
37 1+(−0.161+0.601i)T+(−32.0−18.5i)T2 1 + (-0.161 + 0.601i)T + (-32.0 - 18.5i)T^{2}
41 1−6.44iT−41T2 1 - 6.44iT - 41T^{2}
43 1+(5.47−5.47i)T−43iT2 1 + (5.47 - 5.47i)T - 43iT^{2}
47 1+(5.04+1.35i)T+(40.7+23.5i)T2 1 + (5.04 + 1.35i)T + (40.7 + 23.5i)T^{2}
53 1+(3.87−1.03i)T+(45.8−26.5i)T2 1 + (3.87 - 1.03i)T + (45.8 - 26.5i)T^{2}
59 1+(2.77−4.80i)T+(−29.5−51.0i)T2 1 + (2.77 - 4.80i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.70+6.41i)T+(−30.5+52.8i)T2 1 + (3.70 + 6.41i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.12+1.37i)T+(58.0−33.5i)T2 1 + (-5.12 + 1.37i)T + (58.0 - 33.5i)T^{2}
71 1+3.61iT−71T2 1 + 3.61iT - 71T^{2}
73 1+(2.15+8.05i)T+(−63.2+36.5i)T2 1 + (2.15 + 8.05i)T + (-63.2 + 36.5i)T^{2}
79 1+(−14.7+8.52i)T+(39.5−68.4i)T2 1 + (-14.7 + 8.52i)T + (39.5 - 68.4i)T^{2}
83 1+(−3.21−3.21i)T+83iT2 1 + (-3.21 - 3.21i)T + 83iT^{2}
89 1+(−4.70−8.14i)T+(−44.5+77.0i)T2 1 + (-4.70 - 8.14i)T + (-44.5 + 77.0i)T^{2}
97 1+(−4.39−4.39i)T+97iT2 1 + (-4.39 - 4.39i)T + 97iT^{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

−10.79958357479063302741284375247, −9.228951597639528840019047153226, −8.110789597293399579109261725761, −7.74214610502363585857344106210, −6.66189668944280246182911558317, −5.98484432841818124818915965493, −4.79825380133964109630725755994, −3.75285287907373876619195950761, −3.30788807942689594575852932747, −1.71234318339396330752649275581, 2.08650524078025522169361239509, 3.27287460677019732954224491264, 3.77041814994559503210457929969, 4.60342874043418201147752433483, 5.55246885252355187553896147293, 6.75419005591387220136893255718, 7.54015341062642845736467947848, 8.755422347196559509658886645476, 9.741810216200473448942833050379, 10.87494847407901322895961424838

Graph of the ZZ-function along the critical line