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Properties

Label 1-460-460.267-r1-0-0
Degree $1$
Conductor $460$
Sign $0.859 + 0.510i$
Analytic cond. $49.4338$
Root an. cond. $49.4338$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.989 − 0.142i)3-s + (0.755 − 0.654i)7-s + (0.959 + 0.281i)9-s + (0.841 + 0.540i)11-s + (0.755 + 0.654i)13-s + (0.909 − 0.415i)17-s + (−0.415 + 0.909i)19-s + (−0.841 + 0.540i)21-s + (−0.909 − 0.415i)27-s + (−0.415 − 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.755 − 0.654i)33-s + (−0.281 + 0.959i)37-s + (−0.654 − 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)3-s + (0.755 − 0.654i)7-s + (0.959 + 0.281i)9-s + (0.841 + 0.540i)11-s + (0.755 + 0.654i)13-s + (0.909 − 0.415i)17-s + (−0.415 + 0.909i)19-s + (−0.841 + 0.540i)21-s + (−0.909 − 0.415i)27-s + (−0.415 − 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.755 − 0.654i)33-s + (−0.281 + 0.959i)37-s + (−0.654 − 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.859 + 0.510i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ 0.859 + 0.510i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.671544541 + 0.4591610011i\)
\(L(\frac12)\) \(\approx\) \(1.671544541 + 0.4591610011i\)
\(L(1)\) \(\approx\) \(1.003977794 + 0.03641747952i\)
\(L(1)\) \(\approx\) \(1.003977794 + 0.03641747952i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (0.755 - 0.654i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (0.755 + 0.654i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (-0.281 + 0.959i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.755 - 0.654i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (0.540 + 0.841i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.281 + 0.959i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.69173227575283471513703409532, −22.72064908709729383061974832073, −21.922523579605447852037272843542, −21.32164999205830043044177622768, −20.4105133754586056043682660357, −19.10752015524015452670165761889, −18.4169631259690831144839773251, −17.517270139767402951709932172830, −16.90582253742564641395104722963, −15.86667616126781919144746027325, −15.11640710864520710283504015101, −14.14367601187150891239464674114, −12.90792690218538528069693663959, −12.12508919777799269321519964261, −11.206651448955480195612237570426, −10.70911519569560131412799145069, −9.3936757024528622878570541356, −8.50091774471707751536935023110, −7.36344689926458563646456168055, −6.12092337243185029374397123059, −5.5809471008784594508200454019, −4.48938832181255741575246646471, −3.38266609860414533253043604637, −1.729642475762611205959197480430, −0.661757749884633319520491062041, 1.02812034489258864523666403367, 1.75704818554915945267128579697, 3.77542628945302329484624665877, 4.52740489116915381048045942612, 5.60738632661144497664994185987, 6.62492652104796663777973691469, 7.40245386560372385887848163921, 8.50457970716522091249931249640, 9.85310785260505319319326581242, 10.56433420994596733324480767060, 11.666645405362894804876335722724, 12.01946810402962068329385207828, 13.3093399647872205023489975110, 14.16661582203904492340584396135, 15.09436430899007577592215329019, 16.37386210178220448395892435813, 16.85897054225665819509546354185, 17.68610007487888357056451550506, 18.50597456761433658956266125090, 19.34737692061309926757547990554, 20.64459595896954039598978617165, 21.149261929879672954966332601, 22.227085402842701452541012758174, 23.16011831231239997134953377302, 23.494651759269217062905094168945

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