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Properties

Label 1-460-460.63-r1-0-0
Degree $1$
Conductor $460$
Sign $-0.419 + 0.907i$
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  + (0.540 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.419 + 0.907i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ -0.419 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3119068206 + 0.4876606860i\)
\(L(\frac12)\) \(\approx\) \(0.3119068206 + 0.4876606860i\)
\(L(1)\) \(\approx\) \(0.9626958731 - 0.1231520926i\)
\(L(1)\) \(\approx\) \(0.9626958731 - 0.1231520926i\)

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

−23.35690073543032265928718565786, −22.59283400203959453956604117112, −21.444750310254883827791763519881, −21.026119632750119664970370456964, −20.11508504739180526049587192421, −19.24066790638654916649978029083, −18.55705047254837124158778085777, −16.945172223487513537577621680641, −16.58937395871135030900646805057, −15.761654998778617662928122645476, −14.60912761495067150660199922643, −13.97813564701822934356810794559, −13.22398016004323502197199027493, −11.89562483090069461748453409710, −10.81416821625382131020609833755, −10.11773277591339428714360551482, −9.34441966783176334715705663881, −8.17298874228549588864935099382, −7.47443642258457866072349540214, −6.085649437691676412124575879304, −5.00044504608432502535474972839, −3.89634705858014968060614229567, −3.25441031492388644340812777911, −1.844727040760318538091820664094, −0.13386128408841517719312311191, 1.3744032841997828179903644663, 2.62459155435572426297861437980, 3.19827399426045967859233682194, 4.99269072264077131532645655363, 5.83851619221736306123841153904, 7.05749736369499353625121113993, 7.7645185380719947951586476532, 8.77868459586946281743230729161, 9.58420925959112216381166749525, 10.71977266366248193114747697065, 12.13227421459497321572341680574, 12.52511886121638562848516198985, 13.36227997572792987777119600162, 14.49800984295012267116960227313, 15.19771163477525938583563730577, 16.00669024082170179942105165791, 17.442415843809475765080575488554, 18.09062732059888574043582523150, 18.78178310374449617947822704312, 19.702212039755603969911665226719, 20.405196203116816319335845251800, 21.3456677378844308836203374992, 22.3853617346844619898421059448, 23.220308532461149311709401640793, 24.07067500461349054095514932828

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