↑

Properties

Label 1-3381-3381.113-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.675 - 0.737i$
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  + (−0.0203 + 0.999i)2-s + (−0.999 − 0.0407i)4-s + (0.339 − 0.940i)5-s + (0.0611 − 0.998i)8-s + (0.933 + 0.359i)10-s + (−0.768 + 0.639i)11-s + (0.557 − 0.830i)13-s + (0.996 + 0.0815i)16-s + (−0.999 + 0.0407i)17-s + (0.654 − 0.755i)19-s + (−0.377 + 0.925i)20-s + (−0.623 − 0.781i)22-s + (−0.768 − 0.639i)25-s + (0.818 + 0.574i)26-s + (0.999 − 0.0407i)29-s + ⋯
L(s)  = 1  + (−0.0203 + 0.999i)2-s + (−0.999 − 0.0407i)4-s + (0.339 − 0.940i)5-s + (0.0611 − 0.998i)8-s + (0.933 + 0.359i)10-s + (−0.768 + 0.639i)11-s + (0.557 − 0.830i)13-s + (0.996 + 0.0815i)16-s + (−0.999 + 0.0407i)17-s + (0.654 − 0.755i)19-s + (−0.377 + 0.925i)20-s + (−0.623 − 0.781i)22-s + (−0.768 − 0.639i)25-s + (0.818 + 0.574i)26-s + (0.999 − 0.0407i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.675 - 0.737i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.675 - 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.027844853 - 0.4522952407i\)
\(L(\frac12)\) \(\approx\) \(1.027844853 - 0.4522952407i\)
\(L(1)\) \(\approx\) \(0.9073048184 + 0.1641763235i\)
\(L(1)\) \(\approx\) \(0.9073048184 + 0.1641763235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.0203 + 0.999i)T \)
5 \( 1 + (0.339 - 0.940i)T \)
11 \( 1 + (-0.768 + 0.639i)T \)
13 \( 1 + (0.557 - 0.830i)T \)
17 \( 1 + (-0.999 + 0.0407i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (0.999 - 0.0407i)T \)
31 \( 1 + (-0.959 - 0.281i)T \)
37 \( 1 + (-0.101 + 0.994i)T \)
41 \( 1 + (-0.339 + 0.940i)T \)
43 \( 1 + (0.0611 + 0.998i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (0.685 - 0.728i)T \)
59 \( 1 + (0.933 + 0.359i)T \)
61 \( 1 + (0.818 - 0.574i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (0.452 - 0.891i)T \)
73 \( 1 + (0.262 - 0.965i)T \)
79 \( 1 + (0.142 - 0.989i)T \)
83 \( 1 + (-0.523 + 0.852i)T \)
89 \( 1 + (0.742 - 0.670i)T \)
97 \( 1 + (-0.841 - 0.540i)T \)
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

−18.85745730258803796744637484272, −18.36150573507557257218974333556, −17.87497080049326675220927801485, −17.06773126849338042482528252727, −16.112761640360425291034823403195, −15.46469708438658919216492713830, −14.35256228019788480374343136906, −13.99345325580583677666241029759, −13.41062364811066201241989343969, −12.629696006672199980186700221173, −11.69438918910125037465777678295, −11.206162461110662335247249929975, −10.47401554493310606249710143318, −10.10436406830801236181494367774, −8.99456225110449886676948081649, −8.634968290284091370626875862001, −7.53624650972476268827491141206, −6.79948239972938688733128350153, −5.76568078522445513468528332480, −5.27148790796254281267654707922, −3.98264445855683338838839885594, −3.61710002783099070238663527462, −2.527244106894152142118593437884, −2.141028310119913383915953328821, −1.014221183894514202063247242985, 0.389811156705224941875996214931, 1.34273056904833315027431372750, 2.515964701665608921985488762004, 3.577035625259708650350258772797, 4.71337335355174951453350580888, 4.90282692142625834287366788018, 5.79814155245072710368187455561, 6.49379308056789605104824051993, 7.38162441081128265324999016988, 8.11007181294718234319299395715, 8.656388849164611070127103830889, 9.424968674208165509379705821017, 10.02591731137947186412560746428, 10.87046607500111988118911011761, 11.95451821445183776730197164920, 12.8453550651895218314925089528, 13.23766005386455378941413298566, 13.7169641102574202508988844842, 14.77053715074657739984691101060, 15.4933897092740046507582781678, 15.87158901057661091197440536878, 16.58502619148359477147257582642, 17.372233627567759479142314565489, 18.04016139256373388329504621906, 18.16415012507707976185797446987

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