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Properties

Label 1-2475-2475.502-r0-0-0
Degree $1$
Conductor $2475$
Sign $0.733 - 0.679i$
Analytic cond. $11.4938$
Root an. cond. $11.4938$
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  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (0.406 − 0.913i)7-s + i·8-s + (0.994 − 0.104i)13-s + (0.104 + 0.994i)14-s + (−0.5 − 0.866i)16-s + (0.951 − 0.309i)17-s + 19-s + (0.743 − 0.669i)23-s + (−0.809 + 0.587i)26-s + (−0.587 − 0.809i)28-s + (−0.5 − 0.866i)29-s + (0.913 − 0.406i)31-s + (0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (0.406 − 0.913i)7-s + i·8-s + (0.994 − 0.104i)13-s + (0.104 + 0.994i)14-s + (−0.5 − 0.866i)16-s + (0.951 − 0.309i)17-s + 19-s + (0.743 − 0.669i)23-s + (−0.809 + 0.587i)26-s + (−0.587 − 0.809i)28-s + (−0.5 − 0.866i)29-s + (0.913 − 0.406i)31-s + (0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.733 - 0.679i$
Analytic conductor: \(11.4938\)
Root analytic conductor: \(11.4938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (502, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2475,\ (0:\ ),\ 0.733 - 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.271468262 - 0.4980779040i\)
\(L(\frac12)\) \(\approx\) \(1.271468262 - 0.4980779040i\)
\(L(1)\) \(\approx\) \(0.8933894583 - 0.04874385861i\)
\(L(1)\) \(\approx\) \(0.8933894583 - 0.04874385861i\)

Euler product

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

−19.361029912523273771321115367493, −18.75796952286298201226094388842, −18.28890721311724156632477529239, −17.61406702569140013892393154075, −16.82837736293175654705164409938, −16.07378380703111288121968540634, −15.49521870893774294515562012788, −14.68925622853224996232002728829, −13.695618801906469152364373259617, −12.9117051858135284569575642261, −12.1062952346518990726178646367, −11.5552024766906811335919146229, −10.92926485082371347588456685534, −10.046576891193656101134505056286, −9.30491589113247358745740490507, −8.68349035050632339665059564590, −7.99211723374882007351571442159, −7.28775145641202006067553100763, −6.25896435393550147695065626047, −5.52375486966301728284923302648, −4.49487807910509558908760356583, −3.274527652867557829322643288762, −2.94259155910259260800657821607, −1.59755854681318723257033075155, −1.184836592452396470386939773081, 0.73405887815932826090547024330, 1.26186288657424888909899117312, 2.45532013342403569878825754382, 3.51341206541145302942097928176, 4.50104805845499599405609578966, 5.43521355586566919879820339968, 6.12972096749027758930622967739, 7.06059648951385374372010343987, 7.670661796313083564676089135877, 8.25179129250793368156864635673, 9.18354988417278610883727959221, 9.85342180242111291606898706065, 10.63451380715077137360780615785, 11.18172936992402528557305325909, 11.93091045706811175609588937376, 13.06996852147644144451643706817, 14.002736330763023721984026025629, 14.27697507541257447589578286319, 15.348807625047380687247027007032, 15.9210465870412944250288035072, 16.67334177410618205871639919394, 17.23571400070147466617106329231, 17.87694985564073103677364849100, 18.74779804568218356973448389004, 19.05581212056013851591611733124

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