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Properties

Label 1-460-460.327-r1-0-0
Degree $1$
Conductor $460$
Sign $-0.841 - 0.540i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.841 - 0.540i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ -0.841 - 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3884792194 - 1.323554874i\)
\(L(\frac12)\) \(\approx\) \(0.3884792194 - 1.323554874i\)
\(L(1)\) \(\approx\) \(1.136496288 - 0.3360614781i\)
\(L(1)\) \(\approx\) \(1.136496288 - 0.3360614781i\)

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

−24.418311652626790692628294110892, −23.05817536801079950878742626308, −22.213411244965112358073670813428, −21.58229697908030238980744609166, −20.49110517643895069122949378173, −19.69978615149797862447785898813, −19.225524072218576821503176292965, −18.19090300003417520475044922328, −17.160553820080417901595988110352, −16.11321763347536040902197781103, −15.15308526612514610855779312344, −14.76006139800860761281406110164, −13.602818045878852103234171229988, −12.69739416124313164320077281670, −12.07053045490120252653558596890, −10.53933570368807187363202120104, −9.696330167477671455604072971839, −9.00249535080903229806594931538, −8.06934618611268746386214814566, −7.00976504314719613079170066874, −6.04456745559352806688054562323, −4.64543704972298817429370118127, −3.66395423123233262812249057718, −2.64394605118753053384994060872, −1.68162636279594105054758585876, 0.28657277959117359218781218026, 1.741096953122062192208481223224, 2.90389886819704420097716493102, 3.83708654687332154177252569300, 4.76576165879113590965240061270, 6.65417034781557164420824956067, 6.892205813517366948320737177093, 8.20928006632835137660114338975, 9.20129778356544853992793357553, 9.71341463629887686826391900759, 10.95542194222734251957572214671, 12.02350870160366901166900865684, 13.19796955870357074441997463813, 13.64881099713407172884309970023, 14.605562101275558957299448011783, 15.4296509522950039085278017147, 16.483850639728013097304778075032, 17.1949776224390971352794691587, 18.47691753738005298009735126069, 19.29718168008314550191194417515, 19.87842238755027458913126525498, 20.52656348680230307486763234579, 21.85755544062077570256149316745, 22.1916955925520749788037191251, 23.638077595757345290596491554962

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