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Properties

Label 1-460-460.79-r0-0-0
Degree $1$
Conductor $460$
Sign $-0.898 - 0.438i$
Analytic cond. $2.13623$
Root an. cond. $2.13623$
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  + (0.415 + 0.909i)3-s + (−0.841 − 0.540i)7-s + (−0.654 + 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.841 + 0.540i)13-s + (−0.959 + 0.281i)17-s + (−0.959 − 0.281i)19-s + (0.142 − 0.989i)21-s + (−0.959 − 0.281i)27-s + (−0.959 + 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.841 − 0.540i)33-s + (−0.654 + 0.755i)37-s + (−0.841 − 0.540i)39-s + (−0.654 − 0.755i)41-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)3-s + (−0.841 − 0.540i)7-s + (−0.654 + 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.841 + 0.540i)13-s + (−0.959 + 0.281i)17-s + (−0.959 − 0.281i)19-s + (0.142 − 0.989i)21-s + (−0.959 − 0.281i)27-s + (−0.959 + 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.841 − 0.540i)33-s + (−0.654 + 0.755i)37-s + (−0.841 − 0.540i)39-s + (−0.654 − 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.898 - 0.438i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ -0.898 - 0.438i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02543490913 + 0.1102679551i\)
\(L(\frac12)\) \(\approx\) \(0.02543490913 + 0.1102679551i\)
\(L(1)\) \(\approx\) \(0.7140811403 + 0.1917586002i\)
\(L(1)\) \(\approx\) \(0.7140811403 + 0.1917586002i\)

Euler product

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

−23.38853361991939006150807752561, −22.70318378637170559053130079977, −21.89849051960617710417432602946, −20.59516722587899916925251929024, −19.92228573149395423435685247470, −19.1889578074684142169207241406, −18.36032044310544403205751069665, −17.576950243919107757125760261146, −16.68590812552100652989565243407, −15.24597909893274666248386760300, −14.97442675634528936153560637986, −13.65486845202997918017307080247, −12.737870667903023272702162036418, −12.43882633761988117101243112714, −11.24028235070745809412425796570, −9.88161351587497857929081076048, −9.183753095771099502386136620284, −8.11029551650338413830441454145, −7.16307230867317428980373770191, −6.41595534908984633997274868174, −5.32191772947738144525564912595, −3.916043619389898935841532110012, −2.61158039824647598175217064966, −2.00179681400381309519581265675, −0.05303364817612945202151788562, 2.16198858519279222193348173714, 3.28712579410961880051267330770, 4.093857707566100770618189865684, 5.13696590995330517383934248007, 6.3407012316728374458946591346, 7.348981322467627935795471529862, 8.7013198975692606895937876191, 9.18357825032182624866884270037, 10.41379983915677600070476410579, 10.82091500514089191405944070680, 12.1207252516025923563961641779, 13.35157381341263616968263115489, 13.90771417304929338588619113737, 15.01905837142041991090301349369, 15.72246475196288322542099084210, 16.74957686219742732360557190385, 17.05943333960193908316132134689, 18.671401189477772004894686288896, 19.513665527308520055843087285078, 20.02117837517579509744706318452, 21.10081341307787357126241777251, 21.92031640949967589548152762096, 22.383617242322749741449843100003, 23.59400944172776952314886440732, 24.354061086205409322627797965013

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