↑

Properties

Label 1-555-555.479-r0-0-0
Degree $1$
Conductor $555$
Sign $-0.712 - 0.701i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.984 − 0.173i)2-s + (0.939 + 0.342i)4-s + (−0.766 + 0.642i)7-s + (−0.866 − 0.5i)8-s + (−0.5 + 0.866i)11-s + (−0.342 + 0.939i)13-s + (0.866 − 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (−0.984 + 0.173i)19-s + (0.642 − 0.766i)22-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)26-s + (−0.939 + 0.342i)28-s + (−0.866 − 0.5i)29-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (0.939 + 0.342i)4-s + (−0.766 + 0.642i)7-s + (−0.866 − 0.5i)8-s + (−0.5 + 0.866i)11-s + (−0.342 + 0.939i)13-s + (0.866 − 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (−0.984 + 0.173i)19-s + (0.642 − 0.766i)22-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)26-s + (−0.939 + 0.342i)28-s + (−0.866 − 0.5i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(2.57740\)
Root analytic conductor: \(2.57740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (0:\ ),\ -0.712 - 0.701i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05827071637 - 0.1423262283i\)
\(L(\frac12)\) \(\approx\) \(0.05827071637 - 0.1423262283i\)
\(L(1)\) \(\approx\) \(0.4902867667 + 0.0007914058950i\)
\(L(1)\) \(\approx\) \(0.4902867667 + 0.0007914058950i\)

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 \( 1 + (-0.766 + 0.642i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.342 + 0.939i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (0.642 - 0.766i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + T \)
79 \( 1 + (0.642 + 0.766i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (-0.642 + 0.766i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.75764050903275117470349745222, −23.04546335896120263792437052279, −21.82291140061269791003590231748, −21.02538778391422522235995749300, −19.96088481509187622728026949393, −19.46439274166070205943543869721, −18.71355891604160721224260095992, −17.65150836582454144778839704197, −16.99343496158787375976882654973, −16.23772929813113925518391407277, −15.39327433770166386935026946622, −14.59984743601592431426218522317, −13.22649596594809363209430791999, −12.661097979466333797538205591320, −11.18707450786050367813184526171, −10.62294900113078740840921717777, −9.86373307189199760811612657251, −8.78303712956650457976462092552, −8.04109824842137165914960297682, −7.03593769838472332913589389829, −6.23244947110893544370301665718, −5.22948753859745498538087670940, −3.57526905858859687498121947046, −2.71006731330506159917937253371, −1.245615690600198005831368695926, 0.111945785758281051084694628520, 2.02397465795775948021537248526, 2.58978794536962209034798281960, 3.97848901515356172615258903817, 5.33903748262340845794210403135, 6.618788480134841538903507057191, 7.13070151603753282121685822432, 8.34869638873024939504218743798, 9.265287894077010357516667059118, 9.78837078753681597965786828014, 10.82813419226309986632387677811, 11.81800465326717961697254513890, 12.509403138240043449160377219923, 13.43771852558753858825105188658, 15.009731713943386140049709711712, 15.40712058342468963227310304536, 16.55681800526927601381989379299, 17.024455316390263979870022384582, 18.25279435270695344937653263783, 18.74989584194120805751279263113, 19.483869062751861791079630097636, 20.4695468055995215307164661400, 21.11839619618563861082793618361, 22.09120734720694513455168542797, 22.953610930817968249970725985154

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