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Properties

Label 2-405-9.4-c1-0-1
Degree 22
Conductor 405405
Sign −0.984+0.173i-0.984 + 0.173i
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
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  + (1.36 + 2.36i)2-s + (−2.73 + 4.73i)4-s + (−0.5 + 0.866i)5-s + (0.633 + 1.09i)7-s − 9.46·8-s − 2.73·10-s + (1.13 + 1.96i)11-s + (2.73 − 4.73i)13-s + (−1.73 + 3i)14-s + (−7.46 − 12.9i)16-s + 0.732·17-s − 2.46·19-s + (−2.73 − 4.73i)20-s + (−3.09 + 5.36i)22-s + (−1.73 + 3i)23-s + ⋯
L(s)  = 1  + (0.965 + 1.67i)2-s + (−1.36 + 2.36i)4-s + (−0.223 + 0.387i)5-s + (0.239 + 0.415i)7-s − 3.34·8-s − 0.863·10-s + (0.341 + 0.592i)11-s + (0.757 − 1.31i)13-s + (−0.462 + 0.801i)14-s + (−1.86 − 3.23i)16-s + 0.177·17-s − 0.565·19-s + (−0.610 − 1.05i)20-s + (−0.660 + 1.14i)22-s + (−0.361 + 0.625i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((−0.984+0.173i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((−0.984+0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −0.984+0.173i-0.984 + 0.173i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(271,⋅)\chi_{405} (271, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), −0.984+0.173i)(2,\ 405,\ (\ :1/2),\ -0.984 + 0.173i)

Particular Values

L(1)L(1) ≈\approx 0.165825−1.89539i0.165825 - 1.89539i
L(12)L(\frac12) ≈\approx 0.165825−1.89539i0.165825 - 1.89539i
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)
bad3 1 1
5 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good2 1+(−1.36−2.36i)T+(−1+1.73i)T2 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2}
7 1+(−0.633−1.09i)T+(−3.5+6.06i)T2 1 + (-0.633 - 1.09i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1.13−1.96i)T+(−5.5+9.52i)T2 1 + (-1.13 - 1.96i)T + (-5.5 + 9.52i)T^{2}
13 1+(−2.73+4.73i)T+(−6.5−11.2i)T2 1 + (-2.73 + 4.73i)T + (-6.5 - 11.2i)T^{2}
17 1−0.732T+17T2 1 - 0.732T + 17T^{2}
19 1+2.46T+19T2 1 + 2.46T + 19T^{2}
23 1+(1.73−3i)T+(−11.5−19.9i)T2 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3.59−6.23i)T+(−14.5+25.1i)T2 1 + (-3.59 - 6.23i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.5+2.59i)T+(−15.5−26.8i)T2 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2}
37 1−0.732T+37T2 1 - 0.732T + 37T^{2}
41 1+(1.59−2.76i)T+(−20.5−35.5i)T2 1 + (1.59 - 2.76i)T + (-20.5 - 35.5i)T^{2}
43 1+(−5.09−8.83i)T+(−21.5+37.2i)T2 1 + (-5.09 - 8.83i)T + (-21.5 + 37.2i)T^{2}
47 1+(−2.63−4.56i)T+(−23.5+40.7i)T2 1 + (-2.63 - 4.56i)T + (-23.5 + 40.7i)T^{2}
53 1−3.26T+53T2 1 - 3.26T + 53T^{2}
59 1+(−5.86+10.1i)T+(−29.5−51.0i)T2 1 + (-5.86 + 10.1i)T + (-29.5 - 51.0i)T^{2}
61 1+(2+3.46i)T+(−30.5+52.8i)T2 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.73+3i)T+(−33.5−58.0i)T2 1 + (-1.73 + 3i)T + (-33.5 - 58.0i)T^{2}
71 1+0.267T+71T2 1 + 0.267T + 71T^{2}
73 1−9.66T+73T2 1 - 9.66T + 73T^{2}
79 1+(4.26+7.39i)T+(−39.5+68.4i)T2 1 + (4.26 + 7.39i)T + (-39.5 + 68.4i)T^{2}
83 1+(−4.09−7.09i)T+(−41.5+71.8i)T2 1 + (-4.09 - 7.09i)T + (-41.5 + 71.8i)T^{2}
89 1+5.19T+89T2 1 + 5.19T + 89T^{2}
97 1+(3.83+6.63i)T+(−48.5+84.0i)T2 1 + (3.83 + 6.63i)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

−12.18207642515705751637940758412, −11.02205020827738593807648067126, −9.603806174748282933469960518479, −8.428848140622265578115970261404, −7.899804467210816922651734806742, −6.88765769128456567961206306058, −6.04479816052292695329768466713, −5.19779654704415700788058986464, −4.10165376231819603599411643757, −3.04367738719417493586361704995, 1.01053617757719889937601477821, 2.35090648063785735326048693140, 3.90719652756023183783281195154, 4.28105763848237623438696308070, 5.57693200434407207387072893433, 6.59904456475014027428731226369, 8.488341614347387466734063390495, 9.180599278617522443174533777525, 10.30567940076423927479461632115, 10.94902446248956599800103503468

Graph of the ZZ-function along the critical line