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Properties

Label 2-405-9.7-c3-0-24
Degree 22
Conductor 405405
Sign 0.173−0.984i0.173 - 0.984i
Analytic cond. 23.895723.8957
Root an. cond. 4.888334.88833
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  + (−1.29 + 2.23i)2-s + (0.659 + 1.14i)4-s + (2.5 + 4.33i)5-s + (11.4 − 19.8i)7-s − 24.0·8-s − 12.9·10-s + (5.54 − 9.59i)11-s + (5.81 + 10.0i)13-s + (29.5 + 51.2i)14-s + (25.8 − 44.7i)16-s − 10.0·17-s + 117.·19-s + (−3.29 + 5.70i)20-s + (14.3 + 24.8i)22-s + (86.2 + 149. i)23-s + ⋯
L(s)  = 1  + (−0.456 + 0.791i)2-s + (0.0824 + 0.142i)4-s + (0.223 + 0.387i)5-s + (0.618 − 1.07i)7-s − 1.06·8-s − 0.408·10-s + (0.151 − 0.263i)11-s + (0.124 + 0.215i)13-s + (0.564 + 0.978i)14-s + (0.404 − 0.699i)16-s − 0.143·17-s + 1.42·19-s + (−0.0368 + 0.0638i)20-s + (0.138 + 0.240i)22-s + (0.781 + 1.35i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((0.173−0.984i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+3/2) L(s)=((0.173−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: 0.173−0.984i0.173 - 0.984i
Analytic conductor: 23.895723.8957
Root analytic conductor: 4.888334.88833
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ405(136,⋅)\chi_{405} (136, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :3/2), 0.173−0.984i)(2,\ 405,\ (\ :3/2),\ 0.173 - 0.984i)

Particular Values

L(2)L(2) ≈\approx 1.7629616021.762961602
L(12)L(\frac12) ≈\approx 1.7629616021.762961602
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)
bad3 1 1
5 1+(−2.5−4.33i)T 1 + (-2.5 - 4.33i)T
good2 1+(1.29−2.23i)T+(−4−6.92i)T2 1 + (1.29 - 2.23i)T + (-4 - 6.92i)T^{2}
7 1+(−11.4+19.8i)T+(−171.5−297.i)T2 1 + (-11.4 + 19.8i)T + (-171.5 - 297. i)T^{2}
11 1+(−5.54+9.59i)T+(−665.5−1.15e3i)T2 1 + (-5.54 + 9.59i)T + (-665.5 - 1.15e3i)T^{2}
13 1+(−5.81−10.0i)T+(−1.09e3+1.90e3i)T2 1 + (-5.81 - 10.0i)T + (-1.09e3 + 1.90e3i)T^{2}
17 1+10.0T+4.91e3T2 1 + 10.0T + 4.91e3T^{2}
19 1−117.T+6.85e3T2 1 - 117.T + 6.85e3T^{2}
23 1+(−86.2−149.i)T+(−6.08e3+1.05e4i)T2 1 + (-86.2 - 149. i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(−89.1+154.i)T+(−1.21e4−2.11e4i)T2 1 + (-89.1 + 154. i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+(70.2+121.i)T+(−1.48e4+2.57e4i)T2 1 + (70.2 + 121. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1−250.T+5.06e4T2 1 - 250.T + 5.06e4T^{2}
41 1+(−180.−313.i)T+(−3.44e4+5.96e4i)T2 1 + (-180. - 313. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(−180.+312.i)T+(−3.97e4−6.88e4i)T2 1 + (-180. + 312. i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+(300.−519.i)T+(−5.19e4−8.99e4i)T2 1 + (300. - 519. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+201.T+1.48e5T2 1 + 201.T + 1.48e5T^{2}
59 1+(−207.−360.i)T+(−1.02e5+1.77e5i)T2 1 + (-207. - 360. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−27.3+47.3i)T+(−1.13e5−1.96e5i)T2 1 + (-27.3 + 47.3i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(−265.−459.i)T+(−1.50e5+2.60e5i)T2 1 + (-265. - 459. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−933.T+3.57e5T2 1 - 933.T + 3.57e5T^{2}
73 1+560.T+3.89e5T2 1 + 560.T + 3.89e5T^{2}
79 1+(405.−702.i)T+(−2.46e5−4.26e5i)T2 1 + (405. - 702. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+(−269.+466.i)T+(−2.85e5−4.95e5i)T2 1 + (-269. + 466. i)T + (-2.85e5 - 4.95e5i)T^{2}
89 1+686.T+7.04e5T2 1 + 686.T + 7.04e5T^{2}
97 1+(357.−618.i)T+(−4.56e5−7.90e5i)T2 1 + (357. - 618. i)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

−11.27040484114700649437261921582, −9.875116775668530232380435138556, −9.188615980609972601371980205757, −7.85783933925918910048124937291, −7.52205351350530879758988650098, −6.53829152378334720035292922936, −5.53069238968657897966893684976, −4.09028969771938343144417076991, −2.89355350135783222998598041858, −1.04656788045226044229322834463, 0.898097363904004679357912518585, 2.04837573381708597743935586071, 3.09325254621515075778209379249, 4.89367165732304538371634187047, 5.66442700022471041518212490021, 6.81786012983747723282525547457, 8.266881974465515637421131142219, 8.997957079093037564436216190591, 9.680004430075025725644371110276, 10.71557291909569432062262385286

Graph of the ZZ-function along the critical line