Properties

Label 1-3381-3381.647-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.754 + 0.656i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (0.999 + 0.0135i)2-s + (0.999 + 0.0271i)4-s + (0.288 + 0.957i)5-s + (0.999 + 0.0407i)8-s + (0.275 + 0.961i)10-s + (−0.894 + 0.446i)11-s + (−0.794 + 0.607i)13-s + (0.998 + 0.0543i)16-s + (−0.476 + 0.879i)17-s + (−0.0475 − 0.998i)19-s + (0.262 + 0.965i)20-s + (−0.900 + 0.433i)22-s + (−0.833 + 0.552i)25-s + (−0.802 + 0.596i)26-s + (0.523 + 0.852i)29-s + ⋯
L(s)  = 1  + (0.999 + 0.0135i)2-s + (0.999 + 0.0271i)4-s + (0.288 + 0.957i)5-s + (0.999 + 0.0407i)8-s + (0.275 + 0.961i)10-s + (−0.894 + 0.446i)11-s + (−0.794 + 0.607i)13-s + (0.998 + 0.0543i)16-s + (−0.476 + 0.879i)17-s + (−0.0475 − 0.998i)19-s + (0.262 + 0.965i)20-s + (−0.900 + 0.433i)22-s + (−0.833 + 0.552i)25-s + (−0.802 + 0.596i)26-s + (0.523 + 0.852i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.754 + 0.656i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.754 + 0.656i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8225452403 + 2.199265155i\)
\(L(\frac12)\) \(\approx\) \(0.8225452403 + 2.199265155i\)
\(L(1)\) \(\approx\) \(1.606082643 + 0.6366345895i\)
\(L(1)\) \(\approx\) \(1.606082643 + 0.6366345895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.999 + 0.0135i)T \)
5 \( 1 + (0.288 + 0.957i)T \)
11 \( 1 + (-0.894 + 0.446i)T \)
13 \( 1 + (-0.794 + 0.607i)T \)
17 \( 1 + (-0.476 + 0.879i)T \)
19 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (0.523 + 0.852i)T \)
31 \( 1 + (-0.981 - 0.189i)T \)
37 \( 1 + (0.440 + 0.897i)T \)
41 \( 1 + (0.685 - 0.728i)T \)
43 \( 1 + (-0.999 + 0.0407i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (-0.855 + 0.517i)T \)
59 \( 1 + (-0.275 - 0.961i)T \)
61 \( 1 + (0.115 - 0.993i)T \)
67 \( 1 + (0.723 + 0.690i)T \)
71 \( 1 + (-0.742 + 0.670i)T \)
73 \( 1 + (-0.644 + 0.764i)T \)
79 \( 1 + (-0.995 - 0.0950i)T \)
83 \( 1 + (-0.933 - 0.359i)T \)
89 \( 1 + (-0.0339 + 0.999i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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

−18.5577681815863808090895192588, −17.78124626855503551350757754978, −16.94573347809683977799661797450, −16.28338096897335128764734257239, −15.86725386261198547958468930565, −15.03205435629548681375201915191, −14.30403419427174793215598933959, −13.53818608454392525737470018226, −13.05663233202877065656666670535, −12.40930440032025519620807272146, −11.82414524460277394057242454743, −10.9536855991426193490876672806, −10.19014292912593639185819320924, −9.52884676208977510672367438539, −8.46179709338369703978586861631, −7.78072760429505397867717604671, −7.13379849493235961056094405885, −5.90692187363027953280529704335, −5.60973142909604012836827144499, −4.78856060917420413756519769311, −4.22567206823268516478214657248, −3.12354286108900866168499374844, −2.452821061791187377098299864202, −1.59452751283788546900740271601, −0.40496204890598595132513082300, 1.596347143828778702268034086401, 2.390065805768989184297406628364, 2.85879126294889932036582048497, 3.84309444972982965651866808019, 4.652679350379273929276308298164, 5.3238358060402999388182370143, 6.188184610563703802888578036017, 6.9227383869406524007196984971, 7.3183397920163462362056196281, 8.24638072787911064422394272656, 9.40404217612459740254899930949, 10.19248263108183746079225062131, 10.84137590851302484280600403490, 11.332257071588283530949318936419, 12.27054079227596129756592427934, 12.936033712510810522071032473609, 13.50435223172913693819899607810, 14.396858556084305818717348971537, 14.730442134588261362072572330716, 15.52930069989966608172804359164, 15.97500153584023164566949779497, 17.1586801115438956248043578150, 17.481937923891973056839988745707, 18.55818819196736383255820402161, 19.08054227045251910394492394131

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