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Properties

Label 1-465-465.113-r0-0-0
Degree $1$
Conductor $465$
Sign $0.749 + 0.662i$
Analytic cond. $2.15945$
Root an. cond. $2.15945$
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  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.207 + 0.978i)7-s + (0.951 − 0.309i)8-s + (−0.669 − 0.743i)11-s + (0.406 + 0.913i)13-s + (0.669 − 0.743i)14-s + (−0.809 − 0.587i)16-s + (0.743 + 0.669i)17-s + (−0.913 − 0.406i)19-s + (−0.207 + 0.978i)22-s + (0.951 − 0.309i)23-s + (0.5 − 0.866i)26-s + (−0.994 − 0.104i)28-s + (−0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.207 + 0.978i)7-s + (0.951 − 0.309i)8-s + (−0.669 − 0.743i)11-s + (0.406 + 0.913i)13-s + (0.669 − 0.743i)14-s + (−0.809 − 0.587i)16-s + (0.743 + 0.669i)17-s + (−0.913 − 0.406i)19-s + (−0.207 + 0.978i)22-s + (0.951 − 0.309i)23-s + (0.5 − 0.866i)26-s + (−0.994 − 0.104i)28-s + (−0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(2.15945\)
Root analytic conductor: \(2.15945\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 465,\ (0:\ ),\ 0.749 + 0.662i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7476478077 + 0.2830217082i\)
\(L(\frac12)\) \(\approx\) \(0.7476478077 + 0.2830217082i\)
\(L(1)\) \(\approx\) \(0.7526904191 - 0.04494742137i\)
\(L(1)\) \(\approx\) \(0.7526904191 - 0.04494742137i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + (0.207 + 0.978i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (0.406 + 0.913i)T \)
17 \( 1 + (0.743 + 0.669i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (0.951 - 0.309i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.104 + 0.994i)T \)
43 \( 1 + (-0.406 + 0.913i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (-0.207 + 0.978i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (-0.743 + 0.669i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.994 + 0.104i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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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

−23.7057918014387382922480585396, −23.15920111454811056614839023552, −22.564524036040602515371836253374, −20.821325011024237330185958063068, −20.49323221508264752917151825649, −19.29290647132395394764881609868, −18.56002801383816135513017548020, −17.52225568165133482947803826394, −17.10831952222099629618029070315, −16.034909722177346435879429587340, −15.26783267366022299179281625897, −14.41967334405055658284229593002, −13.50038794358799350973641488944, −12.64805966511082273217572169512, −11.06763210048749335154228383129, −10.39299188147173540526135671801, −9.62776515813042202098606246591, −8.40654125086346468684253863266, −7.58555719625102668717149016276, −6.963984528268775516407760664229, −5.65980793334127001524289079816, −4.86675975502696539107418297696, −3.63755779656570460589134461543, −1.94430036110320695413433552586, −0.58140886992595065873270865050, 1.36814274647778784136190438034, 2.460165643107695481662820079899, 3.40873060782172637742111428207, 4.64820078877491543054713720674, 5.81115219172677741770280738551, 7.0628203606957537426526651922, 8.42021675281640376166133335024, 8.70440060514448997156482754457, 9.8400757137805673217901477790, 10.918200590911466688527636250667, 11.47776795171146972478490673811, 12.55166988971140047546800948544, 13.19849297161240987124104013301, 14.374171217631284361951070495, 15.44719956657862506925935002749, 16.48869603255416800674677590300, 17.15549776047641960428421686720, 18.43032634665427719257859419534, 18.73740041840525020810450726469, 19.50844194984422538320585431882, 20.744368546549101406312201998994, 21.43248538225964756675299251889, 21.7894458554353714540697773849, 23.06087982673959722779235781147, 23.95437506740856273056951938874

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