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Properties

Label 2-460-23.8-c1-0-0
Degree $2$
Conductor $460$
Sign $-0.508 - 0.861i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (1.18 + 2.58i)3-s + (0.654 + 0.755i)5-s + (−1.60 − 1.02i)7-s + (−3.32 + 3.84i)9-s + (0.455 + 3.16i)11-s + (−3.20 + 2.05i)13-s + (−1.18 + 2.58i)15-s + (3.02 − 0.888i)17-s + (0.203 + 0.0596i)19-s + (0.770 − 5.35i)21-s + (4.68 + 1.03i)23-s + (−0.142 + 0.989i)25-s + (−5.68 − 1.66i)27-s + (−4.81 + 1.41i)29-s + (2.57 − 5.62i)31-s + ⋯
L(s)  = 1  + (0.681 + 1.49i)3-s + (0.292 + 0.337i)5-s + (−0.605 − 0.389i)7-s + (−1.10 + 1.28i)9-s + (0.137 + 0.954i)11-s + (−0.888 + 0.571i)13-s + (−0.304 + 0.667i)15-s + (0.733 − 0.215i)17-s + (0.0465 + 0.0136i)19-s + (0.168 − 1.16i)21-s + (0.976 + 0.215i)23-s + (−0.0284 + 0.197i)25-s + (−1.09 − 0.321i)27-s + (−0.893 + 0.262i)29-s + (0.461 − 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.508 - 0.861i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.508 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.795060 + 1.39305i\)
\(L(\frac12)\) \(\approx\) \(0.795060 + 1.39305i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (-4.68 - 1.03i)T \)
good3 \( 1 + (-1.18 - 2.58i)T + (-1.96 + 2.26i)T^{2} \)
7 \( 1 + (1.60 + 1.02i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (-0.455 - 3.16i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (3.20 - 2.05i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (-3.02 + 0.888i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (-0.203 - 0.0596i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (4.81 - 1.41i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-2.57 + 5.62i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-1.32 + 1.52i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (4.97 + 5.74i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-4.18 - 9.16i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + (2.46 + 1.58i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-3.02 + 1.94i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-0.657 + 1.44i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (2.15 - 14.9i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (-1.82 + 12.7i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-11.6 - 3.41i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (-7.25 + 4.66i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-3.94 + 4.55i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-0.324 - 0.710i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-3.73 - 4.31i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.03243477883028824493286419609, −10.17665793610518834194812560838, −9.576690024320334993380537270380, −9.167743659442799921406168786181, −7.72583034955636752416674357223, −6.86194766478440262126167278226, −5.40491822079768308347730515362, −4.44064688000869807613530482550, −3.52822559793342673827787586392, −2.42291906138981236345515070294, 0.954566678458559634755111496035, 2.45558200780583941694976248232, 3.34145109627180917202935822711, 5.31217747940249945150946384128, 6.22058156653596608022092287920, 7.13766387377789597896025426092, 8.016758326955328050724870618686, 8.789331007426238484835524165956, 9.593634208858823895794140101573, 10.80869598592060760084246609232

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