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Properties

Label 2-735-105.89-c1-0-70
Degree 22
Conductor 735735
Sign 0.379−0.925i0.379 - 0.925i
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  + (1.36 − 2.36i)2-s + (−1.5 + 0.866i)3-s + (−2.72 − 4.71i)4-s + (−1.93 − 1.11i)5-s + 4.72i·6-s − 9.40·8-s + (1.5 − 2.59i)9-s + (−5.28 + 3.05i)10-s + (8.16 + 4.71i)12-s + 3.87·15-s + (−7.38 + 12.7i)16-s + (−1.50 + 0.868i)17-s + (−4.09 − 7.08i)18-s + (−0.633 − 0.365i)19-s + 12.1i·20-s + ⋯
L(s)  = 1  + (0.964 − 1.67i)2-s + (−0.866 + 0.499i)3-s + (−1.36 − 2.35i)4-s + (−0.866 − 0.499i)5-s + 1.92i·6-s − 3.32·8-s + (0.5 − 0.866i)9-s + (−1.67 + 0.964i)10-s + (2.35 + 1.36i)12-s + 1.00·15-s + (−1.84 + 3.19i)16-s + (−0.364 + 0.210i)17-s + (−0.964 − 1.67i)18-s + (−0.145 − 0.0838i)19-s + 2.72i·20-s + ⋯

Functional equation

Λ(s)=(735s/2 ΓC(s) L(s)=((0.379−0.925i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(735s/2 ΓC(s+1/2) L(s)=((0.379−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 735735    =    3⋅5⋅723 \cdot 5 \cdot 7^{2}
Sign: 0.379−0.925i0.379 - 0.925i
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), 0.379−0.925i)(2,\ 735,\ (\ :1/2),\ 0.379 - 0.925i)

Particular Values

L(1)L(1) ≈\approx 0.278636+0.186919i0.278636 + 0.186919i
L(12)L(\frac12) ≈\approx 0.278636+0.186919i0.278636 + 0.186919i
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+(−1.36+2.36i)T+(−1−1.73i)T2 1 + (-1.36 + 2.36i)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+(1.50−0.868i)T+(8.5−14.7i)T2 1 + (1.50 - 0.868i)T + (8.5 - 14.7i)T^{2}
19 1+(0.633+0.365i)T+(9.5+16.4i)T2 1 + (0.633 + 0.365i)T + (9.5 + 16.4i)T^{2}
23 1+(3.31−5.73i)T+(−11.5−19.9i)T2 1 + (3.31 - 5.73i)T + (-11.5 - 19.9i)T^{2}
29 1−29T2 1 - 29T^{2}
31 1+(−3.54+2.04i)T+(15.5−26.8i)T2 1 + (-3.54 + 2.04i)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+(11.8+6.83i)T+(23.5+40.7i)T2 1 + (11.8 + 6.83i)T + (23.5 + 40.7i)T^{2}
53 1+(7.25+12.5i)T+(−26.5+45.8i)T2 1 + (7.25 + 12.5i)T + (-26.5 + 45.8i)T^{2}
59 1+(−29.5+51.0i)T2 1 + (-29.5 + 51.0i)T^{2}
61 1+(11.7+6.77i)T+(30.5+52.8i)T2 1 + (11.7 + 6.77i)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+(8.39−14.5i)T+(−39.5−68.4i)T2 1 + (8.39 - 14.5i)T + (-39.5 - 68.4i)T^{2}
83 1−10.1iT−83T2 1 - 10.1iT - 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

−9.949091474941318864866534035801, −9.410556183388722186928516084835, −8.257504797086117390326218640680, −6.61216688123674386066355707851, −5.54322369718546455992688227308, −4.80892248539293258061056788169, −4.04677631204267386943109816537, −3.29815486009201404755093691168, −1.58491331983702887863941245395, −0.14290459423679411181904900616, 2.96228899726449226571025117009, 4.31885424542533392178548343095, 4.80127392638770088870133528630, 6.13265413770993834575311731207, 6.46627550062594913351448772119, 7.43610335996495044186470683685, 7.926001553289077411904792843126, 8.844407678035175978787183421973, 10.36323941919505449438083017669, 11.42540002687409057169397383860

Graph of the ZZ-function along the critical line