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Properties

Label 2-1620-12.11-c1-0-25
Degree 22
Conductor 16201620
Sign −0.997+0.0692i-0.997 + 0.0692i
Analytic cond. 12.935712.9357
Root an. cond. 3.596633.59663
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  + (1.03 + 0.964i)2-s + (0.138 + 1.99i)4-s + i·5-s + 3.54i·7-s + (−1.78 + 2.19i)8-s + (−0.964 + 1.03i)10-s − 4.36·11-s + 4.76·13-s + (−3.42 + 3.66i)14-s + (−3.96 + 0.552i)16-s + 3.02i·17-s + 1.16i·19-s + (−1.99 + 0.138i)20-s + (−4.51 − 4.21i)22-s + 3.11·23-s + ⋯
L(s)  = 1  + (0.731 + 0.682i)2-s + (0.0692 + 0.997i)4-s + 0.447i·5-s + 1.34i·7-s + (−0.629 + 0.776i)8-s + (−0.305 + 0.326i)10-s − 1.31·11-s + 1.32·13-s + (−0.915 + 0.980i)14-s + (−0.990 + 0.138i)16-s + 0.732i·17-s + 0.266i·19-s + (−0.446 + 0.0309i)20-s + (−0.962 − 0.897i)22-s + 0.648·23-s + ⋯

Functional equation

Λ(s)=(1620s/2 ΓC(s) L(s)=((−0.997+0.0692i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0692i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1620s/2 ΓC(s+1/2) L(s)=((−0.997+0.0692i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0692i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16201620    =    22⋅34⋅52^{2} \cdot 3^{4} \cdot 5
Sign: −0.997+0.0692i-0.997 + 0.0692i
Analytic conductor: 12.935712.9357
Root analytic conductor: 3.596633.59663
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1620(971,⋅)\chi_{1620} (971, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1620, ( :1/2), −0.997+0.0692i)(2,\ 1620,\ (\ :1/2),\ -0.997 + 0.0692i)

Particular Values

L(1)L(1) ≈\approx 1.9696615981.969661598
L(12)L(\frac12) ≈\approx 1.9696615981.969661598
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+(−1.03−0.964i)T 1 + (-1.03 - 0.964i)T
3 1 1
5 1−iT 1 - iT
good7 1−3.54iT−7T2 1 - 3.54iT - 7T^{2}
11 1+4.36T+11T2 1 + 4.36T + 11T^{2}
13 1−4.76T+13T2 1 - 4.76T + 13T^{2}
17 1−3.02iT−17T2 1 - 3.02iT - 17T^{2}
19 1−1.16iT−19T2 1 - 1.16iT - 19T^{2}
23 1−3.11T+23T2 1 - 3.11T + 23T^{2}
29 1+8.14iT−29T2 1 + 8.14iT - 29T^{2}
31 1+4.79iT−31T2 1 + 4.79iT - 31T^{2}
37 1+7.83T+37T2 1 + 7.83T + 37T^{2}
41 1+4.61iT−41T2 1 + 4.61iT - 41T^{2}
43 1−8.70iT−43T2 1 - 8.70iT - 43T^{2}
47 1+8.13T+47T2 1 + 8.13T + 47T^{2}
53 1−11.9iT−53T2 1 - 11.9iT - 53T^{2}
59 1−0.684T+59T2 1 - 0.684T + 59T^{2}
61 1−10.5T+61T2 1 - 10.5T + 61T^{2}
67 1−7.95iT−67T2 1 - 7.95iT - 67T^{2}
71 1+10.6T+71T2 1 + 10.6T + 71T^{2}
73 1−9.41T+73T2 1 - 9.41T + 73T^{2}
79 1−8.75iT−79T2 1 - 8.75iT - 79T^{2}
83 1−13.3T+83T2 1 - 13.3T + 83T^{2}
89 1+4.93iT−89T2 1 + 4.93iT - 89T^{2}
97 1+6.64T+97T2 1 + 6.64T + 97T^{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

−9.677876455315487857402296480351, −8.604866152617798420754956887661, −8.242942436194401743760412922531, −7.38457759513873709482168109036, −6.22647109440790276516750513083, −5.87836288668472917435888961268, −5.08397359401234278439439191391, −3.94261860488014295140743007812, −2.98845601000050087076299265952, −2.17153761868542668190897646246, 0.57758421035527287809658599789, 1.66888108450421833263211336018, 3.14265467029319864242349880923, 3.73701674290961530434690049725, 4.96330049431419473957220535531, 5.21729442337423099493757813632, 6.58168035138344284954709411941, 7.17523572855155239703420545397, 8.294799699780709106929396387762, 9.083595364975923842008330159783

Graph of the ZZ-function along the critical line