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Properties

Label 2-459-153.101-c2-0-19
Degree $2$
Conductor $459$
Sign $-0.420 - 0.907i$
Analytic cond. $12.5068$
Root an. cond. $3.53650$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.33 + 1.92i)2-s + (5.41 + 9.38i)4-s + (0.961 + 1.66i)5-s + (1.81 + 1.04i)7-s + 26.3i·8-s + 7.40i·10-s + (8.01 − 13.8i)11-s + (4.87 + 8.44i)13-s + (4.03 + 6.99i)14-s + (−29.0 + 50.3i)16-s + (−16.9 − 1.77i)17-s − 4.56·19-s + (−10.4 + 18.0i)20-s + (53.4 − 30.8i)22-s + (−10.8 − 18.8i)23-s + ⋯
L(s)  = 1  + (1.66 + 0.962i)2-s + (1.35 + 2.34i)4-s + (0.192 + 0.332i)5-s + (0.259 + 0.149i)7-s + 3.29i·8-s + 0.740i·10-s + (0.728 − 1.26i)11-s + (0.375 + 0.649i)13-s + (0.288 + 0.499i)14-s + (−1.81 + 3.14i)16-s + (−0.994 − 0.104i)17-s − 0.240·19-s + (−0.520 + 0.902i)20-s + (2.43 − 1.40i)22-s + (−0.473 − 0.819i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $-0.420 - 0.907i$
Analytic conductor: \(12.5068\)
Root analytic conductor: \(3.53650\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 459,\ (\ :1),\ -0.420 - 0.907i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.801190116\)
\(L(\frac12)\) \(\approx\) \(4.801190116\)
\(L(2)\) 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 \( 1 + (16.9 + 1.77i)T \)
good2 \( 1 + (-3.33 - 1.92i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-0.961 - 1.66i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-1.81 - 1.04i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-8.01 + 13.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-4.87 - 8.44i)T + (-84.5 + 146. i)T^{2} \)
19 \( 1 + 4.56T + 361T^{2} \)
23 \( 1 + (10.8 + 18.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (13.6 - 23.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-7.06 + 4.07i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 59.8iT - 1.36e3T^{2} \)
41 \( 1 + (18.6 + 32.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-31.9 + 55.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-49.9 - 28.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 63.4iT - 2.80e3T^{2} \)
59 \( 1 + (9.56 - 5.51i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-71.3 - 41.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (15.9 + 27.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 108.T + 5.04e3T^{2} \)
73 \( 1 + 9.85iT - 5.32e3T^{2} \)
79 \( 1 + (81.2 + 46.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-10.6 - 6.15i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 105. iT - 7.92e3T^{2} \)
97 \( 1 + (106. + 61.7i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.47234656751875487157790399115, −10.69715426253463852975601794648, −8.821654488108904840943052198384, −8.289179511159629098151877384106, −6.83965789392954695242357174894, −6.47966676420482162524230740328, −5.52159175769126646599893210180, −4.41760291236697567839676560794, −3.56538465675384982417265253199, −2.33244439548167981548646938621, 1.35097625204805718092048023801, 2.38480683024493043692487948366, 3.83011057390089000560391769900, 4.50900548591872762688218272995, 5.50122964313355792148006991673, 6.41241977435697078229772429222, 7.47173764639096336623095004377, 9.190181082410168028816976847361, 9.964578455547148754763223672683, 10.97006509199056200206426154157

Graph of the $Z$-function along the critical line