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Properties

Label 2-460-23.6-c1-0-6
Degree $2$
Conductor $460$
Sign $-0.192 + 0.981i$
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  + (−0.169 − 0.0497i)3-s + (−0.841 + 0.540i)5-s + (0.0167 − 0.116i)7-s + (−2.49 − 1.60i)9-s + (−2.18 − 4.77i)11-s + (−0.469 − 3.26i)13-s + (0.169 − 0.0497i)15-s + (2.47 − 2.85i)17-s + (4.45 + 5.14i)19-s + (−0.00861 + 0.0188i)21-s + (−0.322 − 4.78i)23-s + (0.415 − 0.909i)25-s + (0.690 + 0.796i)27-s + (3.36 − 3.88i)29-s + (−6.09 + 1.79i)31-s + ⋯
L(s)  = 1  + (−0.0978 − 0.0287i)3-s + (−0.376 + 0.241i)5-s + (0.00631 − 0.0439i)7-s + (−0.832 − 0.535i)9-s + (−0.657 − 1.43i)11-s + (−0.130 − 0.906i)13-s + (0.0437 − 0.0128i)15-s + (0.599 − 0.691i)17-s + (1.02 + 1.17i)19-s + (−0.00188 + 0.00411i)21-s + (−0.0671 − 0.997i)23-s + (0.0830 − 0.181i)25-s + (0.132 + 0.153i)27-s + (0.625 − 0.721i)29-s + (−1.09 + 0.321i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.192 + 0.981i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.192 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.550922 - 0.669222i\)
\(L(\frac12)\) \(\approx\) \(0.550922 - 0.669222i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (0.322 + 4.78i)T \)
good3 \( 1 + (0.169 + 0.0497i)T + (2.52 + 1.62i)T^{2} \)
7 \( 1 + (-0.0167 + 0.116i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (2.18 + 4.77i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.469 + 3.26i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-2.47 + 2.85i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (-4.45 - 5.14i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-3.36 + 3.88i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (6.09 - 1.79i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (7.87 + 5.05i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-0.642 + 0.413i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (11.0 + 3.24i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 - 2.37T + 47T^{2} \)
53 \( 1 + (0.334 - 2.32i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-1.13 - 7.92i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (-9.33 + 2.73i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (5.95 - 13.0i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-1.23 + 2.71i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (6.14 + 7.09i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-0.499 - 3.47i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (3.99 + 2.57i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (-12.5 - 3.68i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (-7.96 + 5.11i)T + (40.2 - 88.2i)T^{2} \)
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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

−10.76348219465567372000617296397, −10.12724643493798461489402091149, −8.805493705505597822039067214915, −8.151012800328549221512977212114, −7.21749268910537761359246289112, −5.86923553148467535502093152549, −5.37196446399846947614375420430, −3.58741722358536163279147240966, −2.89360097974560150274609971414, −0.53372043710623074137795942606, 1.90660859688307423380723602743, 3.33800461814675462675703803241, 4.78340769769620821018199206317, 5.36570414843975352396549837609, 6.87979021370849820251360990549, 7.60628661859351290564878499324, 8.597131196586876387457500139910, 9.535167046982695890982247133070, 10.40364921825140299237985796946, 11.45912548009693205512329743164

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