↑

Properties

Label 1-3381-3381.860-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.383 + 0.923i$
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  + (−0.101 + 0.994i)2-s + (−0.979 − 0.202i)4-s + (0.986 + 0.162i)5-s + (0.301 − 0.953i)8-s + (−0.262 + 0.965i)10-s + (−0.947 + 0.320i)11-s + (−0.182 − 0.983i)13-s + (0.917 + 0.396i)16-s + (−0.979 + 0.202i)17-s + (−0.415 + 0.909i)19-s + (−0.933 − 0.359i)20-s + (−0.222 − 0.974i)22-s + (0.947 + 0.320i)25-s + (0.996 − 0.0815i)26-s + (0.979 − 0.202i)29-s + ⋯
L(s)  = 1  + (−0.101 + 0.994i)2-s + (−0.979 − 0.202i)4-s + (0.986 + 0.162i)5-s + (0.301 − 0.953i)8-s + (−0.262 + 0.965i)10-s + (−0.947 + 0.320i)11-s + (−0.182 − 0.983i)13-s + (0.917 + 0.396i)16-s + (−0.979 + 0.202i)17-s + (−0.415 + 0.909i)19-s + (−0.933 − 0.359i)20-s + (−0.222 − 0.974i)22-s + (0.947 + 0.320i)25-s + (0.996 − 0.0815i)26-s + (0.979 − 0.202i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.383 + 0.923i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (860, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.383 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.232511705 + 0.8231056516i\)
\(L(\frac12)\) \(\approx\) \(1.232511705 + 0.8231056516i\)
\(L(1)\) \(\approx\) \(0.9068862677 + 0.4493235310i\)
\(L(1)\) \(\approx\) \(0.9068862677 + 0.4493235310i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.101 + 0.994i)T \)
5 \( 1 + (0.986 + 0.162i)T \)
11 \( 1 + (-0.947 + 0.320i)T \)
13 \( 1 + (-0.182 - 0.983i)T \)
17 \( 1 + (-0.979 + 0.202i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (0.979 - 0.202i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (0.488 - 0.872i)T \)
41 \( 1 + (0.986 + 0.162i)T \)
43 \( 1 + (-0.301 - 0.953i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (0.591 - 0.806i)T \)
59 \( 1 + (0.262 - 0.965i)T \)
61 \( 1 + (-0.996 - 0.0815i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.714 + 0.699i)T \)
73 \( 1 + (-0.970 + 0.242i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (-0.377 - 0.925i)T \)
89 \( 1 + (-0.862 + 0.505i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
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

−18.59692666130992647991597785930, −18.16680629005992298214578949630, −17.4490960723769708109112065572, −16.89039998556040113317215730631, −16.05867582391456560215019868825, −15.128270588342518842556253157870, −14.20651171595719559839858719867, −13.59522599168157711874692430900, −13.18122489995185592152568235684, −12.495858765446436893352138644785, −11.54864027404065693129279465143, −10.97650270274657637832138120643, −10.30621346152740815013294214970, −9.55707184231530572611642157, −8.99801545075035549169620817440, −8.37900894095752625126975778896, −7.354801521408019733206934598649, −6.37283362564272505241888056244, −5.63896569793873456607000683662, −4.55118614060288053644483862730, −4.445776229005305986454958160916, −2.82410748899101772333942826323, −2.57919967630295222317725665536, −1.69665915538594343815687370818, −0.69383467561309283948980635808, 0.67571743722339583184940688522, 1.91481896011970449477280602698, 2.72035059381792186701579432245, 3.80463260165190035964362490488, 4.79857531844365402377493172231, 5.39319345003655918864670636895, 6.034290541581612231368769254446, 6.75198780737067619057050619313, 7.50515274006543724216050420503, 8.31689402006850554131787791060, 8.86235813908390947972207001383, 9.87609176468974395927632805010, 10.281752607111799601794434280957, 10.88724205973511831464484648132, 12.37911993416265983000238787184, 12.8846923026619502550584990928, 13.46524707382740024694396719669, 14.23779905086972897722768928901, 14.78813400273079264583842596668, 15.62365853963543787373035550898, 16.0177770592883561620008566481, 17.126100237631938913776273427024, 17.4221569094810951173386560855, 18.19959569960757271697201637721, 18.48332241435038455241374134965

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