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Properties

Label 2-220-20.3-c1-0-18
Degree 22
Conductor 220220
Sign 0.999−0.0120i0.999 - 0.0120i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 0.00742i)2-s + (0.373 − 0.373i)3-s + (1.99 − 0.0210i)4-s + (1.17 + 1.90i)5-s + (0.525 − 0.530i)6-s + (−2.76 − 2.76i)7-s + (2.82 − 0.0445i)8-s + 2.72i·9-s + (1.68 + 2.67i)10-s + i·11-s + (0.739 − 0.754i)12-s + (−3.64 − 3.64i)13-s + (−3.92 − 3.88i)14-s + (1.14 + 0.269i)15-s + (3.99 − 0.0840i)16-s + (1.81 − 1.81i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.00525i)2-s + (0.215 − 0.215i)3-s + (0.999 − 0.0105i)4-s + (0.527 + 0.849i)5-s + (0.214 − 0.216i)6-s + (−1.04 − 1.04i)7-s + (0.999 − 0.0157i)8-s + 0.907i·9-s + (0.531 + 0.847i)10-s + 0.301i·11-s + (0.213 − 0.217i)12-s + (−1.01 − 1.01i)13-s + (−1.04 − 1.03i)14-s + (0.296 + 0.0695i)15-s + (0.999 − 0.0210i)16-s + (0.440 − 0.440i)17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.999−0.0120i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0120i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.999−0.0120i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0120i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.999−0.0120i0.999 - 0.0120i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(23,⋅)\chi_{220} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.999−0.0120i)(2,\ 220,\ (\ :1/2),\ 0.999 - 0.0120i)

Particular Values

L(1)L(1) ≈\approx 2.22522+0.0134504i2.22522 + 0.0134504i
L(12)L(\frac12) ≈\approx 2.22522+0.0134504i2.22522 + 0.0134504i
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+(−1.41+0.00742i)T 1 + (-1.41 + 0.00742i)T
5 1+(−1.17−1.90i)T 1 + (-1.17 - 1.90i)T
11 1−iT 1 - iT
good3 1+(−0.373+0.373i)T−3iT2 1 + (-0.373 + 0.373i)T - 3iT^{2}
7 1+(2.76+2.76i)T+7iT2 1 + (2.76 + 2.76i)T + 7iT^{2}
13 1+(3.64+3.64i)T+13iT2 1 + (3.64 + 3.64i)T + 13iT^{2}
17 1+(−1.81+1.81i)T−17iT2 1 + (-1.81 + 1.81i)T - 17iT^{2}
19 1+4.29T+19T2 1 + 4.29T + 19T^{2}
23 1+(0.0710−0.0710i)T−23iT2 1 + (0.0710 - 0.0710i)T - 23iT^{2}
29 1−2.00iT−29T2 1 - 2.00iT - 29T^{2}
31 1+5.16iT−31T2 1 + 5.16iT - 31T^{2}
37 1+(−2.56+2.56i)T−37iT2 1 + (-2.56 + 2.56i)T - 37iT^{2}
41 1+5.55T+41T2 1 + 5.55T + 41T^{2}
43 1+(6.95−6.95i)T−43iT2 1 + (6.95 - 6.95i)T - 43iT^{2}
47 1+(−6.82−6.82i)T+47iT2 1 + (-6.82 - 6.82i)T + 47iT^{2}
53 1+(6.31+6.31i)T+53iT2 1 + (6.31 + 6.31i)T + 53iT^{2}
59 1+3.52T+59T2 1 + 3.52T + 59T^{2}
61 1−10.1T+61T2 1 - 10.1T + 61T^{2}
67 1+(−2.03−2.03i)T+67iT2 1 + (-2.03 - 2.03i)T + 67iT^{2}
71 1−2.60iT−71T2 1 - 2.60iT - 71T^{2}
73 1+(−3.23−3.23i)T+73iT2 1 + (-3.23 - 3.23i)T + 73iT^{2}
79 1−5.62T+79T2 1 - 5.62T + 79T^{2}
83 1+(−2.63+2.63i)T−83iT2 1 + (-2.63 + 2.63i)T - 83iT^{2}
89 1+8.67iT−89T2 1 + 8.67iT - 89T^{2}
97 1+(−12.7+12.7i)T−97iT2 1 + (-12.7 + 12.7i)T - 97iT^{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

−12.83185292619047937817718601670, −11.30826384910503044313791277827, −10.32788101153553398960753975789, −9.926491913417958112154603189552, −7.75101834697557697711531475813, −7.12415315126340381503844193903, −6.14827578222632260452669118881, −4.87046505866635540448624480254, −3.39031215097116196672169839289, −2.36641387220949498164034886444, 2.22771130520291193550428187627, 3.60042951001474505129146073287, 4.89901984968193412113485051548, 6.02262806281102327845303017498, 6.72141644350886702871901483088, 8.521210076503837615459004480424, 9.383678070803018891737363864669, 10.23994324965805698465459425390, 11.98266443217945602061223399785, 12.24170551714720859520817218456

Graph of the ZZ-function along the critical line