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Properties

Label 2-460-23.9-c1-0-1
Degree $2$
Conductor $460$
Sign $0.999 + 0.0211i$
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  + (−2.08 − 2.40i)3-s + (0.142 + 0.989i)5-s + (−0.796 + 1.74i)7-s + (−1.01 + 7.04i)9-s + (4.46 + 1.31i)11-s + (0.742 + 1.62i)13-s + (2.08 − 2.40i)15-s + (0.0167 + 0.0107i)17-s + (2.95 − 1.89i)19-s + (5.85 − 1.71i)21-s + (−3.63 − 3.13i)23-s + (−0.959 + 0.281i)25-s + (11.0 − 7.08i)27-s + (2.28 + 1.46i)29-s + (3.56 − 4.11i)31-s + ⋯
L(s)  = 1  + (−1.20 − 1.38i)3-s + (0.0636 + 0.442i)5-s + (−0.300 + 0.658i)7-s + (−0.337 + 2.34i)9-s + (1.34 + 0.395i)11-s + (0.205 + 0.451i)13-s + (0.537 − 0.620i)15-s + (0.00405 + 0.00260i)17-s + (0.676 − 0.435i)19-s + (1.27 − 0.374i)21-s + (−0.756 − 0.653i)23-s + (−0.191 + 0.0563i)25-s + (2.12 − 1.36i)27-s + (0.423 + 0.272i)29-s + (0.639 − 0.738i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.999 + 0.0211i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.999 + 0.0211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.927794 - 0.00983434i\)
\(L(\frac12)\) \(\approx\) \(0.927794 - 0.00983434i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (3.63 + 3.13i)T \)
good3 \( 1 + (2.08 + 2.40i)T + (-0.426 + 2.96i)T^{2} \)
7 \( 1 + (0.796 - 1.74i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-4.46 - 1.31i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-0.742 - 1.62i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-0.0167 - 0.0107i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-2.95 + 1.89i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-2.28 - 1.46i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-3.56 + 4.11i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (1.23 - 8.55i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-1.45 - 10.1i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-1.39 - 1.60i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + (-1.66 + 3.63i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-3.06 - 6.71i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-0.257 + 0.297i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-2.16 + 0.635i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (15.2 - 4.47i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-11.3 + 7.26i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (5.04 + 11.0i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (2.21 - 15.4i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-2.47 - 2.85i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (1.14 + 7.97i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.52935713772694785400819522311, −10.35609925701324283879706764375, −9.271360980399721131022755392554, −8.102135896441942065338265358971, −7.01591813702310014827468943072, −6.44830331545887651803564551736, −5.79292551495055487954589679470, −4.46125282740383748394565189396, −2.57160759638466960861873527776, −1.25420553534103712613383674746, 0.815893642670343575605510523969, 3.66604312884740237296462889618, 4.13482435825074562607221418162, 5.42141228476975512650639101731, 6.03024022496958345733987043063, 7.15944002116600350467714895773, 8.725981287720073585135348851960, 9.488079093344708576443121275815, 10.25280141139589984461448659324, 10.91204445797234898086094660394

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