↑

Properties

Label 2-140-140.23-c1-0-19
Degree 22
Conductor 140140
Sign −0.524+0.851i-0.524 + 0.851i
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  + (1.18 − 0.764i)2-s + (−2.38 − 0.638i)3-s + (0.831 − 1.81i)4-s + (−0.525 − 2.17i)5-s + (−3.32 + 1.06i)6-s + (−2.10 + 1.60i)7-s + (−0.401 − 2.79i)8-s + (2.68 + 1.54i)9-s + (−2.28 − 2.18i)10-s + (4.09 − 2.36i)11-s + (−3.14 + 3.80i)12-s + (0.0592 + 0.0592i)13-s + (−1.27 + 3.51i)14-s + (−0.135 + 5.51i)15-s + (−2.61 − 3.02i)16-s + (4.77 + 1.27i)17-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−1.37 − 0.368i)3-s + (0.415 − 0.909i)4-s + (−0.235 − 0.971i)5-s + (−1.35 + 0.433i)6-s + (−0.794 + 0.607i)7-s + (−0.142 − 0.989i)8-s + (0.893 + 0.515i)9-s + (−0.723 − 0.690i)10-s + (1.23 − 0.713i)11-s + (−0.907 + 1.09i)12-s + (0.0164 + 0.0164i)13-s + (−0.339 + 0.940i)14-s + (−0.0349 + 1.42i)15-s + (−0.654 − 0.755i)16-s + (1.15 + 0.310i)17-s + ⋯

Functional equation

Λ(s)=(140s/2 ΓC(s) L(s)=((−0.524+0.851i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(140s/2 ΓC(s+1/2) L(s)=((−0.524+0.851i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 140140    =    22⋅5⋅72^{2} \cdot 5 \cdot 7
Sign: −0.524+0.851i-0.524 + 0.851i
Analytic conductor: 1.117901.11790
Root analytic conductor: 1.057311.05731
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ140(23,⋅)\chi_{140} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 140, ( :1/2), −0.524+0.851i)(2,\ 140,\ (\ :1/2),\ -0.524 + 0.851i)

Particular Values

L(1)L(1) ≈\approx 0.509780−0.912498i0.509780 - 0.912498i
L(12)L(\frac12) ≈\approx 0.509780−0.912498i0.509780 - 0.912498i
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+(−1.18+0.764i)T 1 + (-1.18 + 0.764i)T
5 1+(0.525+2.17i)T 1 + (0.525 + 2.17i)T
7 1+(2.10−1.60i)T 1 + (2.10 - 1.60i)T
good3 1+(2.38+0.638i)T+(2.59+1.5i)T2 1 + (2.38 + 0.638i)T + (2.59 + 1.5i)T^{2}
11 1+(−4.09+2.36i)T+(5.5−9.52i)T2 1 + (-4.09 + 2.36i)T + (5.5 - 9.52i)T^{2}
13 1+(−0.0592−0.0592i)T+13iT2 1 + (-0.0592 - 0.0592i)T + 13iT^{2}
17 1+(−4.77−1.27i)T+(14.7+8.5i)T2 1 + (-4.77 - 1.27i)T + (14.7 + 8.5i)T^{2}
19 1+(−1.31+2.27i)T+(−9.5−16.4i)T2 1 + (-1.31 + 2.27i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.292−1.09i)T+(−19.9+11.5i)T2 1 + (-0.292 - 1.09i)T + (-19.9 + 11.5i)T^{2}
29 1−7.27iT−29T2 1 - 7.27iT - 29T^{2}
31 1+(−4.01+2.31i)T+(15.5−26.8i)T2 1 + (-4.01 + 2.31i)T + (15.5 - 26.8i)T^{2}
37 1+(0.596+2.22i)T+(−32.0+18.5i)T2 1 + (0.596 + 2.22i)T + (-32.0 + 18.5i)T^{2}
41 1+5.71T+41T2 1 + 5.71T + 41T^{2}
43 1+(−1.57+1.57i)T−43iT2 1 + (-1.57 + 1.57i)T - 43iT^{2}
47 1+(2.67−0.716i)T+(40.7−23.5i)T2 1 + (2.67 - 0.716i)T + (40.7 - 23.5i)T^{2}
53 1+(−2.44+9.12i)T+(−45.8−26.5i)T2 1 + (-2.44 + 9.12i)T + (-45.8 - 26.5i)T^{2}
59 1+(−1.67−2.90i)T+(−29.5+51.0i)T2 1 + (-1.67 - 2.90i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.978−1.69i)T+(−30.5−52.8i)T2 1 + (0.978 - 1.69i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.131+0.491i)T+(−58.0−33.5i)T2 1 + (-0.131 + 0.491i)T + (-58.0 - 33.5i)T^{2}
71 1−14.4iT−71T2 1 - 14.4iT - 71T^{2}
73 1+(2.48−9.26i)T+(−63.2−36.5i)T2 1 + (2.48 - 9.26i)T + (-63.2 - 36.5i)T^{2}
79 1+(3.05−5.28i)T+(−39.5−68.4i)T2 1 + (3.05 - 5.28i)T + (-39.5 - 68.4i)T^{2}
83 1+(−5.62+5.62i)T−83iT2 1 + (-5.62 + 5.62i)T - 83iT^{2}
89 1+(−14.4−8.34i)T+(44.5+77.0i)T2 1 + (-14.4 - 8.34i)T + (44.5 + 77.0i)T^{2}
97 1+(−5.81+5.81i)T−97iT2 1 + (-5.81 + 5.81i)T - 97iT^{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

−12.53276569132532500322316292280, −11.93325933809859549848561200757, −11.35425914435030242314746916764, −9.990358156256052254883390837029, −8.869558512200415988246399431640, −6.84745560435674768508069778518, −5.88753573432637506863298667958, −5.15436931990624916819673430089, −3.59564812655457191389960407807, −1.06837805019699597449568474699, 3.42741919658605946136204007863, 4.53995906911238604223144582898, 6.03369655538813553346183768983, 6.62819917933564965180361005140, 7.62597435546900247736553244837, 9.762013851194959437587344011536, 10.63091056783554984248051287521, 11.88256716248687467482129065845, 12.09241057052902108900440774613, 13.62435895007789363418506979105

Graph of the ZZ-function along the critical line