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Properties

Label 2-1620-9.7-c3-0-9
Degree 22
Conductor 16201620
Sign −0.766−0.642i-0.766 - 0.642i
Analytic cond. 95.583095.5830
Root an. cond. 9.776669.77666
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.5 − 4.33i)5-s + (−11.3 + 19.5i)7-s + (5.48 − 9.50i)11-s + (37.9 + 65.6i)13-s − 15.9·17-s + 58.8·19-s + (−28.3 − 49.0i)23-s + (−12.5 + 21.6i)25-s + (−2.80 + 4.86i)29-s + (60.3 + 104. i)31-s + 113.·35-s + 236.·37-s + (−98.0 − 169. i)41-s + (−9.09 + 15.7i)43-s + (−155. + 270. i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.610 + 1.05i)7-s + (0.150 − 0.260i)11-s + (0.808 + 1.40i)13-s − 0.227·17-s + 0.710·19-s + (−0.256 − 0.444i)23-s + (−0.100 + 0.173i)25-s + (−0.0179 + 0.0311i)29-s + (0.349 + 0.605i)31-s + 0.546·35-s + 1.04·37-s + (−0.373 − 0.647i)41-s + (−0.0322 + 0.0558i)43-s + (−0.483 + 0.838i)47-s + ⋯

Functional equation

Λ(s)=(1620s/2 ΓC(s) L(s)=((−0.766−0.642i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1620s/2 ΓC(s+3/2) L(s)=((−0.766−0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16201620    =    22⋅34⋅52^{2} \cdot 3^{4} \cdot 5
Sign: −0.766−0.642i-0.766 - 0.642i
Analytic conductor: 95.583095.5830
Root analytic conductor: 9.776669.77666
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1620(541,⋅)\chi_{1620} (541, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1620, ( :3/2), −0.766−0.642i)(2,\ 1620,\ (\ :3/2),\ -0.766 - 0.642i)

Particular Values

L(2)L(2) ≈\approx 1.0713096941.071309694
L(12)L(\frac12) ≈\approx 1.0713096941.071309694
L(52)L(\frac{5}{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+(2.5+4.33i)T 1 + (2.5 + 4.33i)T
good7 1+(11.3−19.5i)T+(−171.5−297.i)T2 1 + (11.3 - 19.5i)T + (-171.5 - 297. i)T^{2}
11 1+(−5.48+9.50i)T+(−665.5−1.15e3i)T2 1 + (-5.48 + 9.50i)T + (-665.5 - 1.15e3i)T^{2}
13 1+(−37.9−65.6i)T+(−1.09e3+1.90e3i)T2 1 + (-37.9 - 65.6i)T + (-1.09e3 + 1.90e3i)T^{2}
17 1+15.9T+4.91e3T2 1 + 15.9T + 4.91e3T^{2}
19 1−58.8T+6.85e3T2 1 - 58.8T + 6.85e3T^{2}
23 1+(28.3+49.0i)T+(−6.08e3+1.05e4i)T2 1 + (28.3 + 49.0i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(2.80−4.86i)T+(−1.21e4−2.11e4i)T2 1 + (2.80 - 4.86i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+(−60.3−104.i)T+(−1.48e4+2.57e4i)T2 1 + (-60.3 - 104. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1−236.T+5.06e4T2 1 - 236.T + 5.06e4T^{2}
41 1+(98.0+169.i)T+(−3.44e4+5.96e4i)T2 1 + (98.0 + 169. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(9.09−15.7i)T+(−3.97e4−6.88e4i)T2 1 + (9.09 - 15.7i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+(155.−270.i)T+(−5.19e4−8.99e4i)T2 1 + (155. - 270. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1−33.3T+1.48e5T2 1 - 33.3T + 1.48e5T^{2}
59 1+(260.+451.i)T+(−1.02e5+1.77e5i)T2 1 + (260. + 451. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−75.1+130.i)T+(−1.13e5−1.96e5i)T2 1 + (-75.1 + 130. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(253.+438.i)T+(−1.50e5+2.60e5i)T2 1 + (253. + 438. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−961.T+3.57e5T2 1 - 961.T + 3.57e5T^{2}
73 1+251.T+3.89e5T2 1 + 251.T + 3.89e5T^{2}
79 1+(417.−722.i)T+(−2.46e5−4.26e5i)T2 1 + (417. - 722. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+(345.−598.i)T+(−2.85e5−4.95e5i)T2 1 + (345. - 598. i)T + (-2.85e5 - 4.95e5i)T^{2}
89 1+1.00e3T+7.04e5T2 1 + 1.00e3T + 7.04e5T^{2}
97 1+(642.−1.11e3i)T+(−4.56e5−7.90e5i)T2 1 + (642. - 1.11e3i)T + (-4.56e5 - 7.90e5i)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

−9.207636112411087743972495632279, −8.715721947607096883102270565926, −7.904879381912537539958506740624, −6.70977277628900024237777444623, −6.21577575414845140683503603751, −5.28058387724147035095702971681, −4.31663946479509068026272547726, −3.39858832375687694422553744538, −2.34326440294617030359863329646, −1.20316334822528967394845319659, 0.25984707521403966053905458945, 1.24769914146808889867007860958, 2.82908572314313599213347925021, 3.56206085598588991731149537214, 4.35368256992059494133438665970, 5.56856064245986030874901804803, 6.33264718740796266837256038499, 7.21665043314003426501084571143, 7.78754962661797053564475840894, 8.626551080591181642046947264208

Graph of the ZZ-function along the critical line