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Properties

Label 2-735-105.89-c1-0-52
Degree 22
Conductor 735735
Sign −0.580+0.814i-0.580 + 0.814i
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  + (1.22 − 2.12i)2-s + (−1.5 + 0.866i)3-s + (−2.01 − 3.49i)4-s + (1.93 + 1.11i)5-s + 4.25i·6-s − 4.98·8-s + (1.5 − 2.59i)9-s + (4.75 − 2.74i)10-s + (6.04 + 3.49i)12-s − 3.87·15-s + (−2.09 + 3.62i)16-s + (6.98 − 4.03i)17-s + (−3.68 − 6.38i)18-s + (−4.87 − 2.81i)19-s − 9.01i·20-s + ⋯
L(s)  = 1  + (0.868 − 1.50i)2-s + (−0.866 + 0.499i)3-s + (−1.00 − 1.74i)4-s + (0.866 + 0.499i)5-s + 1.73i·6-s − 1.76·8-s + (0.5 − 0.866i)9-s + (1.50 − 0.868i)10-s + (1.74 + 1.00i)12-s − 1.00·15-s + (−0.523 + 0.907i)16-s + (1.69 − 0.977i)17-s + (−0.868 − 1.50i)18-s + (−1.11 − 0.645i)19-s − 2.01i·20-s + ⋯

Functional equation

Λ(s)=(735s/2 ΓC(s) L(s)=((−0.580+0.814i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(735s/2 ΓC(s+1/2) L(s)=((−0.580+0.814i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 735735    =    3⋅5⋅723 \cdot 5 \cdot 7^{2}
Sign: −0.580+0.814i-0.580 + 0.814i
Analytic conductor: 5.869005.86900
Root analytic conductor: 2.422602.42260
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ735(509,⋅)\chi_{735} (509, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 735, ( :1/2), −0.580+0.814i)(2,\ 735,\ (\ :1/2),\ -0.580 + 0.814i)

Particular Values

L(1)L(1) ≈\approx 0.908986−1.76453i0.908986 - 1.76453i
L(12)L(\frac12) ≈\approx 0.908986−1.76453i0.908986 - 1.76453i
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+(1.5−0.866i)T 1 + (1.5 - 0.866i)T
5 1+(−1.93−1.11i)T 1 + (-1.93 - 1.11i)T
7 1 1
good2 1+(−1.22+2.12i)T+(−1−1.73i)T2 1 + (-1.22 + 2.12i)T + (-1 - 1.73i)T^{2}
11 1+(5.5−9.52i)T2 1 + (5.5 - 9.52i)T^{2}
13 1+13T2 1 + 13T^{2}
17 1+(−6.98+4.03i)T+(8.5−14.7i)T2 1 + (-6.98 + 4.03i)T + (8.5 - 14.7i)T^{2}
19 1+(4.87+2.81i)T+(9.5+16.4i)T2 1 + (4.87 + 2.81i)T + (9.5 + 16.4i)T^{2}
23 1+(−4.79+8.30i)T+(−11.5−19.9i)T2 1 + (-4.79 + 8.30i)T + (-11.5 - 19.9i)T^{2}
29 1−29T2 1 - 29T^{2}
31 1+(3.83−2.21i)T+(15.5−26.8i)T2 1 + (3.83 - 2.21i)T + (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+(−0.886−0.511i)T+(23.5+40.7i)T2 1 + (-0.886 - 0.511i)T + (23.5 + 40.7i)T^{2}
53 1+(−4.71−8.17i)T+(−26.5+45.8i)T2 1 + (-4.71 - 8.17i)T + (-26.5 + 45.8i)T^{2}
59 1+(−29.5+51.0i)T2 1 + (-29.5 + 51.0i)T^{2}
61 1+(3.53+2.04i)T+(30.5+52.8i)T2 1 + (3.53 + 2.04i)T + (30.5 + 52.8i)T^{2}
67 1+(33.5−58.0i)T2 1 + (33.5 - 58.0i)T^{2}
71 1−71T2 1 - 71T^{2}
73 1+(−36.5+63.2i)T2 1 + (-36.5 + 63.2i)T^{2}
79 1+(−2.91+5.05i)T+(−39.5−68.4i)T2 1 + (-2.91 + 5.05i)T + (-39.5 - 68.4i)T^{2}
83 1−15.0iT−83T2 1 - 15.0iT - 83T^{2}
89 1+(−44.5−77.0i)T2 1 + (-44.5 - 77.0i)T^{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

−10.54969362307063072082102444035, −9.701429556246009728346093741217, −8.993672114981928276659238424352, −7.09513752469574518911916901465, −6.11074042885793722850563468179, −5.25457323704571158880267260437, −4.59454772333010287813250008586, −3.38170531237160392159990458876, −2.47235780953709382135403999091, −0.966597498509929579732482621957, 1.59381485743390049544816942015, 3.65949021584265358898443249810, 4.83079270396765441386165308547, 5.70999386846098580632322951646, 5.90511688251882993516150809223, 6.97642298077716148281432396715, 7.76127202010140399448932022227, 8.556283808726046721536824026812, 9.755869848860232466853927010991, 10.66487165883362775612501909145

Graph of the ZZ-function along the critical line