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Properties

Label 2-459-9.7-c1-0-6
Degree 22
Conductor 459459
Sign −0.0947−0.995i-0.0947 - 0.995i
Analytic cond. 3.665133.66513
Root an. cond. 1.914451.91445
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  + (−1.17 + 2.03i)2-s + (−1.76 − 3.06i)4-s + (1.17 + 2.03i)5-s + (1.76 − 3.06i)7-s + 3.61·8-s − 5.53·10-s + (1.80 − 3.12i)11-s + (2.21 + 3.83i)13-s + (4.15 + 7.20i)14-s + (−0.715 + 1.23i)16-s + 17-s + 5.39·19-s + (4.15 − 7.20i)20-s + (4.25 + 7.36i)22-s + (−1.57 − 2.72i)23-s + ⋯
L(s)  = 1  + (−0.831 + 1.44i)2-s + (−0.883 − 1.53i)4-s + (0.526 + 0.911i)5-s + (0.668 − 1.15i)7-s + 1.27·8-s − 1.75·10-s + (0.544 − 0.943i)11-s + (0.614 + 1.06i)13-s + (1.11 + 1.92i)14-s + (−0.178 + 0.309i)16-s + 0.242·17-s + 1.23·19-s + (0.930 − 1.61i)20-s + (0.906 + 1.56i)22-s + (−0.328 − 0.568i)23-s + ⋯

Functional equation

Λ(s)=(459s/2 ΓC(s) L(s)=((−0.0947−0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0947 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(459s/2 ΓC(s+1/2) L(s)=((−0.0947−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0947 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 459459    =    33⋅173^{3} \cdot 17
Sign: −0.0947−0.995i-0.0947 - 0.995i
Analytic conductor: 3.665133.66513
Root analytic conductor: 1.914451.91445
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ459(154,⋅)\chi_{459} (154, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 459, ( :1/2), −0.0947−0.995i)(2,\ 459,\ (\ :1/2),\ -0.0947 - 0.995i)

Particular Values

L(1)L(1) ≈\approx 0.731175+0.804106i0.731175 + 0.804106i
L(12)L(\frac12) ≈\approx 0.731175+0.804106i0.731175 + 0.804106i
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
17 1−T 1 - T
good2 1+(1.17−2.03i)T+(−1−1.73i)T2 1 + (1.17 - 2.03i)T + (-1 - 1.73i)T^{2}
5 1+(−1.17−2.03i)T+(−2.5+4.33i)T2 1 + (-1.17 - 2.03i)T + (-2.5 + 4.33i)T^{2}
7 1+(−1.76+3.06i)T+(−3.5−6.06i)T2 1 + (-1.76 + 3.06i)T + (-3.5 - 6.06i)T^{2}
11 1+(−1.80+3.12i)T+(−5.5−9.52i)T2 1 + (-1.80 + 3.12i)T + (-5.5 - 9.52i)T^{2}
13 1+(−2.21−3.83i)T+(−6.5+11.2i)T2 1 + (-2.21 - 3.83i)T + (-6.5 + 11.2i)T^{2}
19 1−5.39T+19T2 1 - 5.39T + 19T^{2}
23 1+(1.57+2.72i)T+(−11.5+19.9i)T2 1 + (1.57 + 2.72i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.862−1.49i)T+(−14.5−25.1i)T2 1 + (0.862 - 1.49i)T + (-14.5 - 25.1i)T^{2}
31 1+(−3.26−5.66i)T+(−15.5+26.8i)T2 1 + (-3.26 - 5.66i)T + (-15.5 + 26.8i)T^{2}
37 1+11.4T+37T2 1 + 11.4T + 37T^{2}
41 1+(−1.99−3.45i)T+(−20.5+35.5i)T2 1 + (-1.99 - 3.45i)T + (-20.5 + 35.5i)T^{2}
43 1+(−0.907+1.57i)T+(−21.5−37.2i)T2 1 + (-0.907 + 1.57i)T + (-21.5 - 37.2i)T^{2}
47 1+(0.944−1.63i)T+(−23.5−40.7i)T2 1 + (0.944 - 1.63i)T + (-23.5 - 40.7i)T^{2}
53 1+1.55T+53T2 1 + 1.55T + 53T^{2}
59 1+(4.28+7.41i)T+(−29.5+51.0i)T2 1 + (4.28 + 7.41i)T + (-29.5 + 51.0i)T^{2}
61 1+(5.52−9.57i)T+(−30.5−52.8i)T2 1 + (5.52 - 9.57i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.15+3.72i)T+(−33.5+58.0i)T2 1 + (2.15 + 3.72i)T + (-33.5 + 58.0i)T^{2}
71 1−4.67T+71T2 1 - 4.67T + 71T^{2}
73 1−10.6T+73T2 1 - 10.6T + 73T^{2}
79 1+(−2.03+3.53i)T+(−39.5−68.4i)T2 1 + (-2.03 + 3.53i)T + (-39.5 - 68.4i)T^{2}
83 1+(2.22−3.86i)T+(−41.5−71.8i)T2 1 + (2.22 - 3.86i)T + (-41.5 - 71.8i)T^{2}
89 1−12.3T+89T2 1 - 12.3T + 89T^{2}
97 1+(−2.83+4.91i)T+(−48.5−84.0i)T2 1 + (-2.83 + 4.91i)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

−10.88995111735524952322005042105, −10.28211839702647376338995959994, −9.271336361875822180592147926999, −8.456969831317477072143986831595, −7.50633005170683712950714878907, −6.74441658079717991414548004455, −6.17371094755760289925783033110, −4.93171647480171449157683942365, −3.47633934041869629922681002225, −1.24810661714817408080059299928, 1.23449264519445715526081079561, 2.15655325921897698987464211530, 3.50354575943590664500187029090, 4.97273021037963242449202724750, 5.81869841920138663693777044515, 7.69992673568245773291907086562, 8.516115891494018436272279026689, 9.277045773978280310789184572578, 9.774682686288573702715209482327, 10.80193495733503778645791237092

Graph of the ZZ-function along the critical line