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Properties

Label 2-2160-9.4-c1-0-12
Degree 22
Conductor 21602160
Sign 0.710+0.703i0.710 + 0.703i
Analytic cond. 17.247617.2476
Root an. cond. 4.153034.15303
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  + (−0.5 + 0.866i)5-s + (−0.724 − 1.25i)7-s + 2·17-s − 2.89·19-s + (1.27 − 2.20i)23-s + (−0.499 − 0.866i)25-s + (3.94 + 6.84i)29-s + (5.44 − 9.43i)31-s + 1.44·35-s − 6·37-s + (−0.0505 + 0.0874i)41-s + (−3.89 − 6.75i)43-s + (2.27 + 3.94i)47-s + (2.44 − 4.24i)49-s + 11.7·53-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (−0.273 − 0.474i)7-s + 0.485·17-s − 0.665·19-s + (0.265 − 0.460i)23-s + (−0.0999 − 0.173i)25-s + (0.733 + 1.27i)29-s + (0.978 − 1.69i)31-s + 0.245·35-s − 0.986·37-s + (−0.00788 + 0.0136i)41-s + (−0.594 − 1.02i)43-s + (0.331 + 0.574i)47-s + (0.349 − 0.606i)49-s + 1.62·53-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.710+0.703i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1/2) L(s)=((0.710+0.703i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.710+0.703i0.710 + 0.703i
Analytic conductor: 17.247617.2476
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2160(1441,⋅)\chi_{2160} (1441, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1/2), 0.710+0.703i)(2,\ 2160,\ (\ :1/2),\ 0.710 + 0.703i)

Particular Values

L(1)L(1) ≈\approx 1.4499074101.449907410
L(12)L(\frac12) ≈\approx 1.4499074101.449907410
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
3 1 1
5 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good7 1+(0.724+1.25i)T+(−3.5+6.06i)T2 1 + (0.724 + 1.25i)T + (-3.5 + 6.06i)T^{2}
11 1+(−5.5+9.52i)T2 1 + (-5.5 + 9.52i)T^{2}
13 1+(−6.5−11.2i)T2 1 + (-6.5 - 11.2i)T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+2.89T+19T2 1 + 2.89T + 19T^{2}
23 1+(−1.27+2.20i)T+(−11.5−19.9i)T2 1 + (-1.27 + 2.20i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3.94−6.84i)T+(−14.5+25.1i)T2 1 + (-3.94 - 6.84i)T + (-14.5 + 25.1i)T^{2}
31 1+(−5.44+9.43i)T+(−15.5−26.8i)T2 1 + (-5.44 + 9.43i)T + (-15.5 - 26.8i)T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1+(0.0505−0.0874i)T+(−20.5−35.5i)T2 1 + (0.0505 - 0.0874i)T + (-20.5 - 35.5i)T^{2}
43 1+(3.89+6.75i)T+(−21.5+37.2i)T2 1 + (3.89 + 6.75i)T + (-21.5 + 37.2i)T^{2}
47 1+(−2.27−3.94i)T+(−23.5+40.7i)T2 1 + (-2.27 - 3.94i)T + (-23.5 + 40.7i)T^{2}
53 1−11.7T+53T2 1 - 11.7T + 53T^{2}
59 1+(−5.44+9.43i)T+(−29.5−51.0i)T2 1 + (-5.44 + 9.43i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.5−2.59i)T+(−30.5+52.8i)T2 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.62+9.74i)T+(−33.5−58.0i)T2 1 + (-5.62 + 9.74i)T + (-33.5 - 58.0i)T^{2}
71 1+9.79T+71T2 1 + 9.79T + 71T^{2}
73 1+5.79T+73T2 1 + 5.79T + 73T^{2}
79 1+(−1.44−2.51i)T+(−39.5+68.4i)T2 1 + (-1.44 - 2.51i)T + (-39.5 + 68.4i)T^{2}
83 1+(−0.275−0.476i)T+(−41.5+71.8i)T2 1 + (-0.275 - 0.476i)T + (-41.5 + 71.8i)T^{2}
89 1−16.7T+89T2 1 - 16.7T + 89T^{2}
97 1+(1+1.73i)T+(−48.5+84.0i)T2 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{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

−8.853154885102460873745981534711, −8.261032640920457730204564235374, −7.29835036499755280901564035427, −6.75995113304210617087790432692, −5.91736405618493464233945516111, −4.91083687527932679981700741628, −4.00146965476361049211770660167, −3.20147267838816799584546743316, −2.12651015412862400858851335802, −0.61220779306442116758033154828, 1.04409572270346864466305798018, 2.38361779170187557576398274131, 3.35257568424756317001976607024, 4.35414153523709184134041458416, 5.18278829129028159646251404451, 6.01018507917860200364504927600, 6.81289909606171036596486380788, 7.67945889753309376189035334528, 8.591817073897265510153859262402, 8.916266315605884736102548338824

Graph of the ZZ-function along the critical line