↑

Properties

Label 2-140-35.9-c1-0-3
Degree 22
Conductor 140140
Sign 0.669+0.742i0.669 + 0.742i
Analytic cond. 1.117901.11790
Root an. cond. 1.057311.05731
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 0.866i)3-s + (−0.5 − 2.17i)5-s + (−1.13 − 2.38i)7-s + (2.63 + 4.56i)11-s + 2.62i·13-s + (−2.63 − 2.83i)15-s + (−0.362 + 0.209i)17-s + (1.63 − 2.83i)19-s + (−3.77 − 2.59i)21-s + (6.77 + 3.91i)23-s + (−4.50 + 2.17i)25-s + 5.19i·27-s − 4.27·29-s + (−1.63 − 2.83i)31-s + (7.91 + 4.56i)33-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.223 − 0.974i)5-s + (−0.429 − 0.902i)7-s + (0.795 + 1.37i)11-s + 0.728i·13-s + (−0.680 − 0.732i)15-s + (−0.0879 + 0.0507i)17-s + (0.375 − 0.650i)19-s + (−0.823 − 0.566i)21-s + (1.41 + 0.815i)23-s + (−0.900 + 0.435i)25-s + 0.999i·27-s − 0.793·29-s + (−0.294 − 0.509i)31-s + (1.37 + 0.795i)33-s + ⋯

Functional equation

Λ(s)=(140s/2 ΓC(s) L(s)=((0.669+0.742i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(140s/2 ΓC(s+1/2) L(s)=((0.669+0.742i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 140140    =    22⋅5⋅72^{2} \cdot 5 \cdot 7
Sign: 0.669+0.742i0.669 + 0.742i
Analytic conductor: 1.117901.11790
Root analytic conductor: 1.057311.05731
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ140(9,⋅)\chi_{140} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 140, ( :1/2), 0.669+0.742i)(2,\ 140,\ (\ :1/2),\ 0.669 + 0.742i)

Particular Values

L(1)L(1) ≈\approx 1.20551−0.536547i1.20551 - 0.536547i
L(12)L(\frac12) ≈\approx 1.20551−0.536547i1.20551 - 0.536547i
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)
bad2 1 1
5 1+(0.5+2.17i)T 1 + (0.5 + 2.17i)T
7 1+(1.13+2.38i)T 1 + (1.13 + 2.38i)T
good3 1+(−1.5+0.866i)T+(1.5−2.59i)T2 1 + (-1.5 + 0.866i)T + (1.5 - 2.59i)T^{2}
11 1+(−2.63−4.56i)T+(−5.5+9.52i)T2 1 + (-2.63 - 4.56i)T + (-5.5 + 9.52i)T^{2}
13 1−2.62iT−13T2 1 - 2.62iT - 13T^{2}
17 1+(0.362−0.209i)T+(8.5−14.7i)T2 1 + (0.362 - 0.209i)T + (8.5 - 14.7i)T^{2}
19 1+(−1.63+2.83i)T+(−9.5−16.4i)T2 1 + (-1.63 + 2.83i)T + (-9.5 - 16.4i)T^{2}
23 1+(−6.77−3.91i)T+(11.5+19.9i)T2 1 + (-6.77 - 3.91i)T + (11.5 + 19.9i)T^{2}
29 1+4.27T+29T2 1 + 4.27T + 29T^{2}
31 1+(1.63+2.83i)T+(−15.5+26.8i)T2 1 + (1.63 + 2.83i)T + (-15.5 + 26.8i)T^{2}
37 1+(8.63+4.98i)T+(18.5+32.0i)T2 1 + (8.63 + 4.98i)T + (18.5 + 32.0i)T^{2}
41 1+3.72T+41T2 1 + 3.72T + 41T^{2}
43 1+2.15iT−43T2 1 + 2.15iT - 43T^{2}
47 1+(−5.63−3.25i)T+(23.5+40.7i)T2 1 + (-5.63 - 3.25i)T + (23.5 + 40.7i)T^{2}
53 1+(4.91−2.83i)T+(26.5−45.8i)T2 1 + (4.91 - 2.83i)T + (26.5 - 45.8i)T^{2}
59 1+(1.63+2.83i)T+(−29.5+51.0i)T2 1 + (1.63 + 2.83i)T + (-29.5 + 51.0i)T^{2}
61 1+(−6.77+11.7i)T+(−30.5−52.8i)T2 1 + (-6.77 + 11.7i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.04−1.76i)T+(33.5−58.0i)T2 1 + (3.04 - 1.76i)T + (33.5 - 58.0i)T^{2}
71 1+4.54T+71T2 1 + 4.54T + 71T^{2}
73 1+(−5.63+3.25i)T+(36.5−63.2i)T2 1 + (-5.63 + 3.25i)T + (36.5 - 63.2i)T^{2}
79 1+(−3.63+6.30i)T+(−39.5−68.4i)T2 1 + (-3.63 + 6.30i)T + (-39.5 - 68.4i)T^{2}
83 1+7.40iT−83T2 1 + 7.40iT - 83T^{2}
89 1+(3.5−6.06i)T+(−44.5−77.0i)T2 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2}
97 1−6.92iT−97T2 1 - 6.92iT - 97T^{2}
show more
show less
   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

−13.13874524576967597113312175109, −12.29280575824357261179668269929, −11.09122549690287877617055865588, −9.486777281139055447548499103507, −9.016022891953113342712542735998, −7.56724516418755785646199133188, −6.97272893606860122951055659982, −4.94560266360483320586645260055, −3.71715217461558743430330436490, −1.70187967642289420603536752132, 2.92974880914809004783737323761, 3.55780134359169229725739791947, 5.67768649321110556795598673553, 6.78512934521630292500291983834, 8.352237345153777076379897922362, 9.006782490216052697291878239909, 10.14226387607678464297435538674, 11.21232054841931837271673296354, 12.19584353081240943480137896528, 13.55106033424159937871462411124

Graph of the ZZ-function along the critical line