↑

Properties

Label 2-220-20.7-c1-0-14
Degree 22
Conductor 220220
Sign 0.321−0.947i0.321 - 0.947i
Analytic cond. 1.756701.75670
Root an. cond. 1.325401.32540
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  + (1.23 + 0.682i)2-s + (1.16 + 1.16i)3-s + (1.06 + 1.68i)4-s + (−2.21 + 0.280i)5-s + (0.649 + 2.24i)6-s + (2.18 − 2.18i)7-s + (0.172 + 2.82i)8-s − 0.277i·9-s + (−2.93 − 1.16i)10-s − i·11-s + (−0.723 + 3.21i)12-s + (−2.68 + 2.68i)13-s + (4.18 − 1.21i)14-s + (−2.91 − 2.26i)15-s + (−1.71 + 3.61i)16-s + (−1.20 − 1.20i)17-s + ⋯
L(s)  = 1  + (0.876 + 0.482i)2-s + (0.673 + 0.673i)3-s + (0.534 + 0.844i)4-s + (−0.992 + 0.125i)5-s + (0.265 + 0.914i)6-s + (0.824 − 0.824i)7-s + (0.0610 + 0.998i)8-s − 0.0924i·9-s + (−0.929 − 0.368i)10-s − 0.301i·11-s + (−0.208 + 0.929i)12-s + (−0.745 + 0.745i)13-s + (1.11 − 0.324i)14-s + (−0.752 − 0.583i)15-s + (−0.427 + 0.903i)16-s + (−0.292 − 0.292i)17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.321−0.947i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.321−0.947i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.321−0.947i0.321 - 0.947i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(67,⋅)\chi_{220} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.321−0.947i)(2,\ 220,\ (\ :1/2),\ 0.321 - 0.947i)

Particular Values

L(1)L(1) ≈\approx 1.72547+1.23681i1.72547 + 1.23681i
L(12)L(\frac12) ≈\approx 1.72547+1.23681i1.72547 + 1.23681i
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+(−1.23−0.682i)T 1 + (-1.23 - 0.682i)T
5 1+(2.21−0.280i)T 1 + (2.21 - 0.280i)T
11 1+iT 1 + iT
good3 1+(−1.16−1.16i)T+3iT2 1 + (-1.16 - 1.16i)T + 3iT^{2}
7 1+(−2.18+2.18i)T−7iT2 1 + (-2.18 + 2.18i)T - 7iT^{2}
13 1+(2.68−2.68i)T−13iT2 1 + (2.68 - 2.68i)T - 13iT^{2}
17 1+(1.20+1.20i)T+17iT2 1 + (1.20 + 1.20i)T + 17iT^{2}
19 1−0.758T+19T2 1 - 0.758T + 19T^{2}
23 1+(4.06+4.06i)T+23iT2 1 + (4.06 + 4.06i)T + 23iT^{2}
29 1+3.39iT−29T2 1 + 3.39iT - 29T^{2}
31 1+4.55iT−31T2 1 + 4.55iT - 31T^{2}
37 1+(−4.42−4.42i)T+37iT2 1 + (-4.42 - 4.42i)T + 37iT^{2}
41 1−1.63T+41T2 1 - 1.63T + 41T^{2}
43 1+(−7.31−7.31i)T+43iT2 1 + (-7.31 - 7.31i)T + 43iT^{2}
47 1+(1.24−1.24i)T−47iT2 1 + (1.24 - 1.24i)T - 47iT^{2}
53 1+(4.89−4.89i)T−53iT2 1 + (4.89 - 4.89i)T - 53iT^{2}
59 1+8.66T+59T2 1 + 8.66T + 59T^{2}
61 1+13.4T+61T2 1 + 13.4T + 61T^{2}
67 1+(−10.3+10.3i)T−67iT2 1 + (-10.3 + 10.3i)T - 67iT^{2}
71 1−0.631iT−71T2 1 - 0.631iT - 71T^{2}
73 1+(11.0−11.0i)T−73iT2 1 + (11.0 - 11.0i)T - 73iT^{2}
79 1+9.62T+79T2 1 + 9.62T + 79T^{2}
83 1+(−4.44−4.44i)T+83iT2 1 + (-4.44 - 4.44i)T + 83iT^{2}
89 1−6.02iT−89T2 1 - 6.02iT - 89T^{2}
97 1+(1.24+1.24i)T+97iT2 1 + (1.24 + 1.24i)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

−12.45597682988815867395551828428, −11.60142811270599466463207280708, −10.82783327272404292809593572631, −9.416253796777751984029712880556, −8.166073742704267102526673186592, −7.58768300052529649837752582690, −6.38520159324684617276495880516, −4.44637461820754411506830024386, −4.27939123671738021527583071921, −2.84349855274072701476169157303, 1.86347670643013342259963302416, 3.08362205554137600776640548247, 4.55917655560923689657806459914, 5.56214843689261253523277643300, 7.22916211218784803523804399086, 7.913699485798925077394548497853, 9.011149713307147463122804768737, 10.49338528944013501659930069706, 11.44598149318954798607452722178, 12.31405577996967752973389479414

Graph of the ZZ-function along the critical line