Properties

Label 1-460-460.167-r0-0-0
Degree $1$
Conductor $460$
Sign $-0.546 + 0.837i$
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  + (−0.540 + 0.841i)3-s + (−0.281 + 0.959i)7-s + (−0.415 − 0.909i)9-s + (0.654 − 0.755i)11-s + (0.281 + 0.959i)13-s + (0.989 − 0.142i)17-s + (−0.142 + 0.989i)19-s + (−0.654 − 0.755i)21-s + (0.989 + 0.142i)27-s + (0.142 + 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.281 + 0.959i)33-s + (−0.909 + 0.415i)37-s + (−0.959 − 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)3-s + (−0.281 + 0.959i)7-s + (−0.415 − 0.909i)9-s + (0.654 − 0.755i)11-s + (0.281 + 0.959i)13-s + (0.989 − 0.142i)17-s + (−0.142 + 0.989i)19-s + (−0.654 − 0.755i)21-s + (0.989 + 0.142i)27-s + (0.142 + 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.281 + 0.959i)33-s + (−0.909 + 0.415i)37-s + (−0.959 − 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.546 + 0.837i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ -0.546 + 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4481289438 + 0.8279899438i\)
\(L(\frac12)\) \(\approx\) \(0.4481289438 + 0.8279899438i\)
\(L(1)\) \(\approx\) \(0.7713804654 + 0.4012398567i\)
\(L(1)\) \(\approx\) \(0.7713804654 + 0.4012398567i\)

Euler product

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

−23.55539366795401490731129538753, −22.83445161290166261952796390185, −22.33259589901041222256977557279, −20.978043056240426892765201836265, −19.97947352418840196610025922309, −19.46182785630357501743835208578, −18.40359909069960224702254557807, −17.448721027910741950295185904766, −17.065617094426837720873346791163, −16.01930613952677383800860507103, −14.856151707762813510844806743256, −13.85577329562585663771047556613, −13.05190817279728499956428878507, −12.35810815926708352609417750342, −11.3217056674659898128409939664, −10.482512580613864494587891211556, −9.532695214137653634387628155412, −8.10501773725053129793834706038, −7.34753141176729938430531622517, −6.55556895556000554869900622565, −5.5499452675177219315176584088, −4.40177010674660511099905075321, −3.1684914446843456810666571062, −1.74374249219523286843659659781, −0.60354667116656753767360994398, 1.48150780169409042387961770204, 3.15141304621305400744424238025, 3.90586357426315231042160040636, 5.229000915596590666003853189304, 5.92130131834341942757542703326, 6.830903628512045650947044640634, 8.48821339397436989964461636642, 9.122614155628260386339797330535, 10.01826229571480191365264121241, 11.05429039596564414581199050152, 11.89894642327953092159315375458, 12.49573388669612687360781953630, 14.0903102206139359877497054611, 14.64697795484555600926036765044, 15.79263474308000378270115121897, 16.413061534841614951357822744886, 17.0440352939485897990708279841, 18.35927332426209293983254379696, 18.89488055084901290076031256896, 20.025940918784789181696546719117, 21.21495392222705779307980265677, 21.55137597159770109513576820064, 22.41129017249103147098462539707, 23.2204684779293498576854151966, 24.11425516964484703350711778537

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