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Properties

Label 1-555-555.392-r1-0-0
Degree $1$
Conductor $555$
Sign $0.844 - 0.534i$
Analytic cond. $59.6430$
Root an. cond. $59.6430$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.342 − 0.939i)7-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + (−0.642 − 0.766i)19-s + (0.939 − 0.342i)22-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)26-s + (0.984 + 0.173i)28-s + (0.866 + 0.5i)29-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.342 − 0.939i)7-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + (−0.642 − 0.766i)19-s + (0.939 − 0.342i)22-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)26-s + (0.984 + 0.173i)28-s + (0.866 + 0.5i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.844 - 0.534i$
Analytic conductor: \(59.6430\)
Root analytic conductor: \(59.6430\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (392, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (1:\ ),\ 0.844 - 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.229639778 - 0.3564472874i\)
\(L(\frac12)\) \(\approx\) \(1.229639778 - 0.3564472874i\)
\(L(1)\) \(\approx\) \(0.7634461993 - 0.1998882181i\)
\(L(1)\) \(\approx\) \(0.7634461993 - 0.1998882181i\)

Euler product

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

−23.40618078706330778307045404396, −22.54268245376778477043830296781, −21.38341964025817602362997190522, −20.74391472320992586531216424650, −19.48098755476766519950678872989, −18.928010240363616616052395683078, −18.12295637317180550555827041919, −17.40805281024887113174683509185, −16.44074520391818224605771737476, −15.6434625528733944650877896511, −14.94527680340098245180848572050, −14.16585176311344088350627721073, −12.955473966525201714493229383209, −12.003740937621832048800600130929, −10.68478781579610790402426823493, −10.40848552672075894843808104848, −8.91874304848806986013445760001, −8.416278394577478753441573211004, −7.693694006402493762891469578898, −6.23445735893577682752207461010, −5.77956141205140706242938004601, −4.74170049082243063411295775754, −3.08925562381248808982363464689, −1.93792000364502583072533535618, −0.645902449412350402383867615262, 0.70442565121995416803700104462, 1.80595985957437992205680534864, 2.868719421415264393214289332604, 4.13624063488678193435594088475, 4.859743009580782974883394693217, 6.731140492915807373016358028420, 7.32183310228177344885248567486, 8.23861942296214534440823605385, 9.38029381376432550765099938800, 9.96579322773143545809998291493, 11.08309084479623252016903708612, 11.53016779770558035183353755630, 12.75357230913091515526542515115, 13.45926862862327695750428165268, 14.46517709918822855935751458541, 15.72225192344997760134437399751, 16.49165828348930812080130285837, 17.42926234060279746639086822243, 17.91119137062383844507924438612, 18.97167198542547808088723165189, 19.684272622276191665926548781135, 20.5953723753068077135764725552, 21.046975086691441989949314900096, 22.00498831979487104977647178460, 23.1057245746882158021565302574

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