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Properties

Label 2-459-9.7-c1-0-6
Degree $2$
Conductor $459$
Sign $-0.0947 - 0.995i$
Analytic cond. $3.66513$
Root an. cond. $1.91445$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $-0.0947 - 0.995i$
Analytic conductor: \(3.66513\)
Root analytic conductor: \(1.91445\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 459,\ (\ :1/2),\ -0.0947 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.731175 + 0.804106i\)
\(L(\frac12)\) \(\approx\) \(0.731175 + 0.804106i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 - T \)
good2 \( 1 + (1.17 - 2.03i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.17 - 2.03i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.76 + 3.06i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.80 + 3.12i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.21 - 3.83i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 - 5.39T + 19T^{2} \)
23 \( 1 + (1.57 + 2.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.862 - 1.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.26 - 5.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 + (-1.99 - 3.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.907 + 1.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.944 - 1.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.55T + 53T^{2} \)
59 \( 1 + (4.28 + 7.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.52 - 9.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.15 + 3.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.67T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + (-2.03 + 3.53i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.22 - 3.86i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + (-2.83 + 4.91i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line