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Properties

Label 2-735-105.59-c1-0-58
Degree 22
Conductor 735735
Sign −0.803+0.595i-0.803 + 0.595i
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  + (0.306 + 1.70i)3-s + (1 − 1.73i)4-s + (−1.93 + 1.11i)5-s + (−2.81 + 1.04i)9-s + (−5.12 − 2.95i)11-s + (3.25 + 1.17i)12-s − 2.64·13-s + (−2.5 − 2.95i)15-s + (−1.99 − 3.46i)16-s + (−1.93 − 1.11i)17-s + 4.47i·20-s + (2.5 − 4.33i)25-s + (−2.64 − 4.47i)27-s + 5.91i·29-s + (3.47 − 9.64i)33-s + ⋯
L(s)  = 1  + (0.177 + 0.984i)3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.499i)5-s + (−0.937 + 0.348i)9-s + (−1.54 − 0.891i)11-s + (0.940 + 0.338i)12-s − 0.733·13-s + (−0.645 − 0.763i)15-s + (−0.499 − 0.866i)16-s + (−0.469 − 0.271i)17-s + 0.999i·20-s + (0.5 − 0.866i)25-s + (−0.509 − 0.860i)27-s + 1.09i·29-s + (0.604 − 1.67i)33-s + ⋯

Functional equation

Λ(s)=(735s/2 ΓC(s) L(s)=((−0.803+0.595i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(735s/2 ΓC(s+1/2) L(s)=((−0.803+0.595i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 735735    =    3⋅5⋅723 \cdot 5 \cdot 7^{2}
Sign: −0.803+0.595i-0.803 + 0.595i
Analytic conductor: 5.869005.86900
Root analytic conductor: 2.422602.42260
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ735(374,⋅)\chi_{735} (374, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 735, ( :1/2), −0.803+0.595i)(2,\ 735,\ (\ :1/2),\ -0.803 + 0.595i)

Particular Values

L(1)L(1) ≈\approx 0.0446230−0.135039i0.0446230 - 0.135039i
L(12)L(\frac12) ≈\approx 0.0446230−0.135039i0.0446230 - 0.135039i
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+(−0.306−1.70i)T 1 + (-0.306 - 1.70i)T
5 1+(1.93−1.11i)T 1 + (1.93 - 1.11i)T
7 1 1
good2 1+(−1+1.73i)T2 1 + (-1 + 1.73i)T^{2}
11 1+(5.12+2.95i)T+(5.5+9.52i)T2 1 + (5.12 + 2.95i)T + (5.5 + 9.52i)T^{2}
13 1+2.64T+13T2 1 + 2.64T + 13T^{2}
17 1+(1.93+1.11i)T+(8.5+14.7i)T2 1 + (1.93 + 1.11i)T + (8.5 + 14.7i)T^{2}
19 1+(9.5−16.4i)T2 1 + (9.5 - 16.4i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1−5.91iT−29T2 1 - 5.91iT - 29T^{2}
31 1+(15.5+26.8i)T2 1 + (15.5 + 26.8i)T^{2}
37 1+(18.5−32.0i)T2 1 + (18.5 - 32.0i)T^{2}
41 1+41T2 1 + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1+(9.68−5.59i)T+(23.5−40.7i)T2 1 + (9.68 - 5.59i)T + (23.5 - 40.7i)T^{2}
53 1+(−26.5−45.8i)T2 1 + (-26.5 - 45.8i)T^{2}
59 1+(−29.5−51.0i)T2 1 + (-29.5 - 51.0i)T^{2}
61 1+(30.5−52.8i)T2 1 + (30.5 - 52.8i)T^{2}
67 1+(33.5+58.0i)T2 1 + (33.5 + 58.0i)T^{2}
71 1+11.8iT−71T2 1 + 11.8iT - 71T^{2}
73 1+(−5.29+9.16i)T+(−36.5−63.2i)T2 1 + (-5.29 + 9.16i)T + (-36.5 - 63.2i)T^{2}
79 1+(−0.5−0.866i)T+(−39.5+68.4i)T2 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2}
83 1−8.94iT−83T2 1 - 8.94iT - 83T^{2}
89 1+(−44.5+77.0i)T2 1 + (-44.5 + 77.0i)T^{2}
97 1+18.5T+97T2 1 + 18.5T + 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

−10.25576850371377866625268741556, −9.368380969914482382456523676034, −8.302766408860814553068562162393, −7.53032192361578198664318962599, −6.44150101718114467927977264428, −5.33646793979217160602002521253, −4.71468423707253193836834271767, −3.27903497850911129168240966286, −2.53000351921565958319880799780, −0.06376460010592685334312796048, 2.06705041109976935303399422597, 2.94492298740847056535524040493, 4.24060394100272543928750123009, 5.33188276100516328644173972485, 6.72813367137609255600527696043, 7.42290806189520344017294218687, 7.994140838189322223460339747997, 8.554238566861824412403679581572, 9.826117936275046439169283269037, 11.02376377795217713132717321566

Graph of the ZZ-function along the critical line