Properties

Label 1-3381-3381.1142-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.939 - 0.341i$
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  + (−0.742 − 0.670i)2-s + (0.101 + 0.994i)4-s + (0.996 − 0.0815i)5-s + (0.591 − 0.806i)8-s + (−0.794 − 0.607i)10-s + (0.986 + 0.162i)11-s + (−0.768 + 0.639i)13-s + (−0.979 + 0.202i)16-s + (0.101 − 0.994i)17-s + (−0.841 − 0.540i)19-s + (0.182 + 0.983i)20-s + (−0.623 − 0.781i)22-s + (0.986 − 0.162i)25-s + (0.999 + 0.0407i)26-s + (−0.101 + 0.994i)29-s + ⋯
L(s)  = 1  + (−0.742 − 0.670i)2-s + (0.101 + 0.994i)4-s + (0.996 − 0.0815i)5-s + (0.591 − 0.806i)8-s + (−0.794 − 0.607i)10-s + (0.986 + 0.162i)11-s + (−0.768 + 0.639i)13-s + (−0.979 + 0.202i)16-s + (0.101 − 0.994i)17-s + (−0.841 − 0.540i)19-s + (0.182 + 0.983i)20-s + (−0.623 − 0.781i)22-s + (0.986 − 0.162i)25-s + (0.999 + 0.0407i)26-s + (−0.101 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.939 - 0.341i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.939 - 0.341i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.406631238 - 0.2478688327i\)
\(L(\frac12)\) \(\approx\) \(1.406631238 - 0.2478688327i\)
\(L(1)\) \(\approx\) \(0.9066852500 - 0.1989420698i\)
\(L(1)\) \(\approx\) \(0.9066852500 - 0.1989420698i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.742 - 0.670i)T \)
5 \( 1 + (0.996 - 0.0815i)T \)
11 \( 1 + (0.986 + 0.162i)T \)
13 \( 1 + (-0.768 + 0.639i)T \)
17 \( 1 + (0.101 - 0.994i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (-0.101 + 0.994i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + (0.862 + 0.505i)T \)
41 \( 1 + (-0.996 + 0.0815i)T \)
43 \( 1 + (0.591 + 0.806i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (-0.452 + 0.891i)T \)
59 \( 1 + (-0.794 - 0.607i)T \)
61 \( 1 + (0.999 - 0.0407i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
71 \( 1 + (0.377 + 0.925i)T \)
73 \( 1 + (-0.992 - 0.122i)T \)
79 \( 1 + (-0.415 + 0.909i)T \)
83 \( 1 + (0.557 + 0.830i)T \)
89 \( 1 + (0.262 - 0.965i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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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

−18.830590118893253190477844544710, −17.97787639555277696090530542792, −17.22853984520591709154657716894, −17.108110984489152881621769438831, −16.34865028494337876038791084106, −15.32262324559545995587049183329, −14.69512748815340296131167444439, −14.329314427178888646061815036120, −13.44411955621124394962535922858, −12.677919327470234141144962872718, −11.825251396327837048197787055525, −10.73545029452425513161473166391, −10.34379332803030261539419497743, −9.620062439347724917892186161939, −8.96195978131879802392726317678, −8.30804669612356568769014833073, −7.45605387364219713504144041972, −6.64476136527050220172826931723, −6.03862836379174571264630921739, −5.50142456756732906567610132519, −4.56070485296560871137045472501, −3.54550775039635622885861934241, −2.23991819749252134232449534867, −1.76872855422569871634504362633, −0.69492087943382189436018717118, 0.8264003378835093546298593536, 1.72120899045284917886752540555, 2.37249443299768780739994315036, 3.12633544264001955598392981885, 4.263788781798715973597429491144, 4.8299527369849564835915293683, 5.99781226860372474826853140176, 6.85182287463502094280892555113, 7.28426340369407803128505910444, 8.43678889738570860241686463886, 9.20050455692746265281405243449, 9.49949597761272753677138736382, 10.178979546451307840696847430022, 11.14118002391163335981577911007, 11.58504434925355158229616225793, 12.55418555265873368124756376255, 12.94480763099944721527494828073, 13.97146627014792120362001107363, 14.38615162514655679877593845128, 15.39994136756298505218725310733, 16.45281461651160428564332331459, 16.92687620535201547119976350446, 17.35080476922408396562095434550, 18.16769894093116532978598328353, 18.728451443672758592726252315154

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