↑

Properties

Label 1-3381-3381.41-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.426 + 0.904i$
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  + (0.714 − 0.699i)2-s + (0.0203 − 0.999i)4-s + (−0.818 + 0.574i)5-s + (−0.685 − 0.728i)8-s + (−0.182 + 0.983i)10-s + (−0.339 − 0.940i)11-s + (−0.882 + 0.470i)13-s + (−0.999 − 0.0407i)16-s + (0.0203 + 0.999i)17-s + (−0.415 − 0.909i)19-s + (0.557 + 0.830i)20-s + (−0.900 − 0.433i)22-s + (0.339 − 0.940i)25-s + (−0.301 + 0.953i)26-s + (−0.0203 − 0.999i)29-s + ⋯
L(s)  = 1  + (0.714 − 0.699i)2-s + (0.0203 − 0.999i)4-s + (−0.818 + 0.574i)5-s + (−0.685 − 0.728i)8-s + (−0.182 + 0.983i)10-s + (−0.339 − 0.940i)11-s + (−0.882 + 0.470i)13-s + (−0.999 − 0.0407i)16-s + (0.0203 + 0.999i)17-s + (−0.415 − 0.909i)19-s + (0.557 + 0.830i)20-s + (−0.900 − 0.433i)22-s + (0.339 − 0.940i)25-s + (−0.301 + 0.953i)26-s + (−0.0203 − 0.999i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.426 + 0.904i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.426 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3809523505 + 0.2416323860i\)
\(L(\frac12)\) \(\approx\) \(0.3809523505 + 0.2416323860i\)
\(L(1)\) \(\approx\) \(0.9150424050 - 0.3975227846i\)
\(L(1)\) \(\approx\) \(0.9150424050 - 0.3975227846i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.714 - 0.699i)T \)
5 \( 1 + (-0.818 + 0.574i)T \)
11 \( 1 + (-0.339 - 0.940i)T \)
13 \( 1 + (-0.882 + 0.470i)T \)
17 \( 1 + (0.0203 + 0.999i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
29 \( 1 + (-0.0203 - 0.999i)T \)
31 \( 1 + (0.142 - 0.989i)T \)
37 \( 1 + (0.742 - 0.670i)T \)
41 \( 1 + (-0.818 + 0.574i)T \)
43 \( 1 + (0.685 - 0.728i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (-0.917 + 0.396i)T \)
59 \( 1 + (0.182 - 0.983i)T \)
61 \( 1 + (0.301 + 0.953i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (0.523 + 0.852i)T \)
73 \( 1 + (-0.794 + 0.607i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (0.488 + 0.872i)T \)
89 \( 1 + (-0.933 + 0.359i)T \)
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

−18.524646919205874480460611251772, −17.875662262421988099760454544113, −17.08965150370531514781858553483, −16.48961309625563244208326782835, −15.87667281306550279042992297752, −15.218456518247036874363236242493, −14.68422646922507189062402598274, −13.970744671926976133239037387565, −12.97885778097377323772718486116, −12.48829751894442986209521633165, −12.05844588505031914598566705842, −11.259617949924362864214851647427, −10.21966190311768479923345227325, −9.38834872184059102520106093591, −8.50861989907269739828062767147, −7.8711828098914945333919002290, −7.272155973010089162074339838807, −6.67668449937133322819201991159, −5.4835297928536863882483988629, −4.92297206576000840652997294976, −4.44114273207128366499054372382, −3.4607532489990399252163488752, −2.776454287489156850027368472614, −1.6657058777904460938049411237, −0.11530474659222543090593514829, 0.94308645725555076886107001118, 2.32960584751952971705773595690, 2.68466425384872057793759115309, 3.75201309300939772901228997985, 4.20441191055632792927123368098, 5.057082069880557893340025944927, 6.03073847091725639410164203643, 6.57393010002609620620411182155, 7.52283526045576805115608377571, 8.28352765302091960348474042715, 9.21423609077516610907042016879, 10.0336981677147205094051566320, 10.769177889669691211741768494740, 11.34918741590278271035852639149, 11.797785624227598556242245612369, 12.748007038939753984949310129285, 13.22581227506758003393882295437, 14.18995966395589537275202717985, 14.638300016458290800461345743874, 15.40031772731932003608881250793, 15.85336522299360599449405853241, 16.86326941537628813748398692556, 17.63994328206575061235019401131, 18.73620366194500380806921507937, 19.05173215166048232981703743608

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