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Properties

Label 1-3381-3381.359-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.902 + 0.431i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.248 − 0.968i)2-s + (−0.876 − 0.482i)4-s + (−0.0882 + 0.996i)5-s + (−0.685 + 0.728i)8-s + (0.942 + 0.333i)10-s + (0.644 + 0.764i)11-s + (0.882 + 0.470i)13-s + (0.534 + 0.844i)16-s + (0.855 + 0.517i)17-s + (0.995 − 0.0950i)19-s + (0.557 − 0.830i)20-s + (0.900 − 0.433i)22-s + (−0.984 − 0.175i)25-s + (0.675 − 0.737i)26-s + (−0.0203 + 0.999i)29-s + ⋯
L(s)  = 1  + (0.248 − 0.968i)2-s + (−0.876 − 0.482i)4-s + (−0.0882 + 0.996i)5-s + (−0.685 + 0.728i)8-s + (0.942 + 0.333i)10-s + (0.644 + 0.764i)11-s + (0.882 + 0.470i)13-s + (0.534 + 0.844i)16-s + (0.855 + 0.517i)17-s + (0.995 − 0.0950i)19-s + (0.557 − 0.830i)20-s + (0.900 − 0.433i)22-s + (−0.984 − 0.175i)25-s + (0.675 − 0.737i)26-s + (−0.0203 + 0.999i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.902 + 0.431i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.902 + 0.431i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.890407552 + 0.4286478243i\)
\(L(\frac12)\) \(\approx\) \(1.890407552 + 0.4286478243i\)
\(L(1)\) \(\approx\) \(1.215692934 - 0.1671474319i\)
\(L(1)\) \(\approx\) \(1.215692934 - 0.1671474319i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.248 - 0.968i)T \)
5 \( 1 + (-0.0882 + 0.996i)T \)
11 \( 1 + (0.644 + 0.764i)T \)
13 \( 1 + (0.882 + 0.470i)T \)
17 \( 1 + (0.855 + 0.517i)T \)
19 \( 1 + (0.995 - 0.0950i)T \)
29 \( 1 + (-0.0203 + 0.999i)T \)
31 \( 1 + (0.928 + 0.371i)T \)
37 \( 1 + (-0.209 + 0.977i)T \)
41 \( 1 + (0.818 + 0.574i)T \)
43 \( 1 + (-0.685 - 0.728i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (-0.802 + 0.596i)T \)
59 \( 1 + (0.942 + 0.333i)T \)
61 \( 1 + (-0.976 + 0.215i)T \)
67 \( 1 + (-0.0475 - 0.998i)T \)
71 \( 1 + (0.523 - 0.852i)T \)
73 \( 1 + (-0.923 + 0.384i)T \)
79 \( 1 + (-0.981 - 0.189i)T \)
83 \( 1 + (0.488 - 0.872i)T \)
89 \( 1 + (0.155 + 0.987i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.7458513211442058856526287421, −17.78952534134350543452839285800, −17.30059813028605143381906180610, −16.52705369731531890068596345443, −15.98302218116170847362900241491, −15.6429294894529323262678361924, −14.527177411940202351191938134992, −13.93315025213280987021414567912, −13.38355251836398430364208670676, −12.67533249734272256595251228808, −11.90523268512691759965616974685, −11.33308230425387736446432402681, −10.02153044466529862500561589681, −9.37338947732194605938870114857, −8.739579175273467074999300586875, −8.02119401458742795614135279368, −7.552367136862324819951665039152, −6.426779903680468555406122700416, −5.74906781327214421243638521986, −5.30294189967598198688223936289, −4.294501088511625184269088569016, −3.71039115334980195901522405863, −2.86828500417514791969664039455, −1.235504939343141894875852556993, −0.64360856171484203680078236578, 1.17859027283730886105081569629, 1.7449506222805337113605231949, 2.85613879992423773454711871463, 3.421621523820065360795857450591, 4.10691147091503224537855436636, 4.97908527145962589321514672210, 5.92875926209024000099524621997, 6.59998949191916045139405112118, 7.45260279238176195905321748846, 8.38226848880666721464003585755, 9.213244538857339916268082818470, 9.92652721105446164242285484706, 10.49001172883289989852172507901, 11.19370409583292420041064213132, 11.93984124282316926489169070999, 12.29070731626345493722474206729, 13.44510744694973072240514470529, 13.90622867793136626121131758616, 14.60170270070045474523146074990, 15.12989716557185398736211411747, 15.99800103218040013368795100972, 17.02394782315492286339089310470, 17.74501276973855966395552133790, 18.40903201126274701685603696919, 18.83846429526580995279954344002

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