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Properties

Label 2-735-105.104-c1-0-70
Degree 22
Conductor 735735
Sign 0.156+0.987i0.156 + 0.987i
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  + 2.45·2-s − 1.73i·3-s + 4.03·4-s − 2.23i·5-s − 4.25i·6-s + 4.98·8-s − 2.99·9-s − 5.49i·10-s − 6.98i·12-s − 3.87·15-s + 4.19·16-s + 8.06i·17-s − 7.36·18-s − 5.62i·19-s − 9.01i·20-s + ⋯
L(s)  = 1  + 1.73·2-s − 0.999i·3-s + 2.01·4-s − 0.999i·5-s − 1.73i·6-s + 1.76·8-s − 0.999·9-s − 1.73i·10-s − 2.01i·12-s − 0.999·15-s + 1.04·16-s + 1.95i·17-s − 1.73·18-s − 1.29i·19-s − 2.01i·20-s + ⋯

Functional equation

Λ(s)=(735s/2 ΓC(s) L(s)=((0.156+0.987i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(735s/2 ΓC(s+1/2) L(s)=((0.156+0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 735735    =    3⋅5⋅723 \cdot 5 \cdot 7^{2}
Sign: 0.156+0.987i0.156 + 0.987i
Analytic conductor: 5.869005.86900
Root analytic conductor: 2.422602.42260
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ735(734,⋅)\chi_{735} (734, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 735, ( :1/2), 0.156+0.987i)(2,\ 735,\ (\ :1/2),\ 0.156 + 0.987i)

Particular Values

L(1)L(1) ≈\approx 3.01520−2.57490i3.01520 - 2.57490i
L(12)L(\frac12) ≈\approx 3.01520−2.57490i3.01520 - 2.57490i
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+1.73iT 1 + 1.73iT
5 1+2.23iT 1 + 2.23iT
7 1 1
good2 1−2.45T+2T2 1 - 2.45T + 2T^{2}
11 1−11T2 1 - 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1−8.06iT−17T2 1 - 8.06iT - 17T^{2}
19 1+5.62iT−19T2 1 + 5.62iT - 19T^{2}
23 1−9.58T+23T2 1 - 9.58T + 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1−4.42iT−31T2 1 - 4.42iT - 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1+1.02iT−47T2 1 + 1.02iT - 47T^{2}
53 1−9.43T+53T2 1 - 9.43T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+4.08iT−61T2 1 + 4.08iT - 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1−71T2 1 - 71T^{2}
73 1+73T2 1 + 73T^{2}
79 1+5.83T+79T2 1 + 5.83T + 79T^{2}
83 1−15.0iT−83T2 1 - 15.0iT - 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+97T2 1 + 97T^{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

−10.72403285965587659356639900631, −9.055162184233632104065484299155, −8.342608783501796109676806698376, −7.17466154433091162502011989568, −6.48158104637942069324271231277, −5.53792746586298195211712858449, −4.88113362675234728932986694625, −3.76215833450480333689431179457, −2.59862615173252316792534236377, −1.35360987229339559928335704982, 2.58860331521543202201731583219, 3.20709784012133919765093998174, 4.13523522116466720489947671982, 5.08862613055917404078844906831, 5.78034352333377268780883799186, 6.78650340920078416595711722034, 7.56653485620405067707913036115, 9.075637560150727912544095930668, 10.01772450706269686019815765952, 10.86909383120797948597968176213

Graph of the ZZ-function along the critical line