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Properties

Label 1-555-555.62-r0-0-0
Degree $1$
Conductor $555$
Sign $0.917 + 0.398i$
Analytic cond. $2.57740$
Root an. cond. $2.57740$
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  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (0.642 + 0.766i)7-s + (0.866 + 0.5i)8-s + (0.5 − 0.866i)11-s + (0.342 − 0.939i)13-s + (0.5 + 0.866i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (0.173 + 0.984i)19-s + (0.642 − 0.766i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + (0.342 + 0.939i)28-s + (0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (0.642 + 0.766i)7-s + (0.866 + 0.5i)8-s + (0.5 − 0.866i)11-s + (0.342 − 0.939i)13-s + (0.5 + 0.866i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (0.173 + 0.984i)19-s + (0.642 − 0.766i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + (0.342 + 0.939i)28-s + (0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.917 + 0.398i$
Analytic conductor: \(2.57740\)
Root analytic conductor: \(2.57740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (0:\ ),\ 0.917 + 0.398i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.854412481 + 0.5925346027i\)
\(L(\frac12)\) \(\approx\) \(2.854412481 + 0.5925346027i\)
\(L(1)\) \(\approx\) \(2.063872073 + 0.2971014736i\)
\(L(1)\) \(\approx\) \(2.063872073 + 0.2971014736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.984 + 0.173i)T \)
7 \( 1 + (0.642 + 0.766i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.342 - 0.939i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.642 + 0.766i)T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.342 - 0.939i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−23.44576398440914190150604659895, −22.36307185399738897623884053829, −21.710338484183456519421804031367, −20.85344711293242087284110978193, −20.027537661975492606177122578598, −19.55943188306987475011215652112, −18.21650828661527868551442815247, −17.27301888781063983602592917685, −16.43795414295068301847486919882, −15.48820452592234686102927594364, −14.51778720133595467158458413510, −14.07192522873652992923902681673, −13.06088167462918614347650209012, −12.23632355136834060553387474703, −11.28119546480845323665483910354, −10.666658819185238709579700661030, −9.58624521500524177360082086227, −8.31917421095284240476594989232, −7.078073185968873996752680295741, −6.618186535544862603357174875939, −5.2760098932544438318210314107, −4.31876733245794273626420645681, −3.80053677066778744520038319131, −2.22068740782064605396549944590, −1.41663963534082917291468638121, 1.46134544705057450589996644055, 2.67251781728966813633626683846, 3.58575241562612213242006256191, 4.70612071913684993229489524599, 5.71757666514741910086082394920, 6.204711508001286168888989366144, 7.67282520115059371941479012190, 8.24904830425450433893911350885, 9.49863497553812117522058357536, 10.84169603499676720306338433555, 11.53545577939437617284441853908, 12.25540996837589374031537996339, 13.24982195351413548925197455326, 14.13061468842601620350845719012, 14.74180076993663556469568912595, 15.787210190151229056628932968584, 16.26475052775804746091072622199, 17.490248214185485920906531091, 18.28057565190688452825456568280, 19.3804921243477528123143808333, 20.32997691266350219367754739909, 21.038066127954797940374046041497, 21.83641640152664548508049434405, 22.50181569491664673335622388318, 23.3151429369581121393198599091

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