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Properties

Label 1-3381-3381.149-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.300 - 0.953i$
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  + (−0.855 + 0.517i)2-s + (0.464 − 0.885i)4-s + (0.644 − 0.764i)5-s + (0.0611 + 0.998i)8-s + (−0.155 + 0.987i)10-s + (0.938 − 0.346i)11-s + (0.557 + 0.830i)13-s + (−0.568 − 0.822i)16-s + (0.534 − 0.844i)17-s + (0.327 − 0.945i)19-s + (−0.377 − 0.925i)20-s + (−0.623 + 0.781i)22-s + (−0.169 − 0.985i)25-s + (−0.906 − 0.421i)26-s + (0.999 + 0.0407i)29-s + ⋯
L(s)  = 1  + (−0.855 + 0.517i)2-s + (0.464 − 0.885i)4-s + (0.644 − 0.764i)5-s + (0.0611 + 0.998i)8-s + (−0.155 + 0.987i)10-s + (0.938 − 0.346i)11-s + (0.557 + 0.830i)13-s + (−0.568 − 0.822i)16-s + (0.534 − 0.844i)17-s + (0.327 − 0.945i)19-s + (−0.377 − 0.925i)20-s + (−0.623 + 0.781i)22-s + (−0.169 − 0.985i)25-s + (−0.906 − 0.421i)26-s + (0.999 + 0.0407i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.300 - 0.953i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.300 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104797592 - 0.8100258597i\)
\(L(\frac12)\) \(\approx\) \(1.104797592 - 0.8100258597i\)
\(L(1)\) \(\approx\) \(0.9030159846 - 0.1094865829i\)
\(L(1)\) \(\approx\) \(0.9030159846 - 0.1094865829i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.855 + 0.517i)T \)
5 \( 1 + (0.644 - 0.764i)T \)
11 \( 1 + (0.938 - 0.346i)T \)
13 \( 1 + (0.557 + 0.830i)T \)
17 \( 1 + (0.534 - 0.844i)T \)
19 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (0.999 + 0.0407i)T \)
31 \( 1 + (0.235 - 0.971i)T \)
37 \( 1 + (-0.810 + 0.585i)T \)
41 \( 1 + (-0.339 - 0.940i)T \)
43 \( 1 + (0.0611 - 0.998i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (-0.973 + 0.229i)T \)
59 \( 1 + (-0.155 + 0.987i)T \)
61 \( 1 + (0.0882 - 0.996i)T \)
67 \( 1 + (-0.580 - 0.814i)T \)
71 \( 1 + (0.452 + 0.891i)T \)
73 \( 1 + (-0.966 - 0.255i)T \)
79 \( 1 + (0.786 - 0.618i)T \)
83 \( 1 + (-0.523 - 0.852i)T \)
89 \( 1 + (0.209 - 0.977i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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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

−19.0624787961706131841609868047, −18.03778199739582662258002770359, −17.836502022263849226280409098033, −17.1349145013294097897668442882, −16.37889131775016862031294577875, −15.64540945894524656898060222789, −14.72363311063466126056478284223, −14.2125435340878065405712461460, −13.25959550087799103409553359078, −12.52157845417608323054030593329, −11.91794780130996787725364949825, −11.03225138342403160514282659233, −10.44079075566160861294618091562, −9.908547354986488842820274747535, −9.26942061076276855394409985221, −8.29051708275637762494521311073, −7.831558405648976361070485112150, −6.72611962137369636083797994131, −6.370063640414049926113683195730, −5.43296469417298292994202148307, −4.14169834104671819722130866948, −3.332630440140229237461414737329, −2.83721292576448222045994611595, −1.56352456313369366066835781734, −1.32347693868062345595522189735, 0.568154796614227788952321171595, 1.348309750914152377531035356879, 2.08315627215596677529180371521, 3.175621412256873631018195134137, 4.41427564876137238050830015143, 5.09688477247181367057043914544, 5.90816227800236868489023330549, 6.58257127946523059916279781822, 7.18906029200955202515512155294, 8.26452746386073534751734415156, 8.84971585029812172027590941194, 9.349404654589260334594421394525, 9.94806547797356113661600356001, 10.850671108121506934131513436493, 11.72122112572492452403555001961, 12.100612840040571618398994132632, 13.51699681644974057415012523920, 13.804001585939118851775835450658, 14.49324272900636941307289633053, 15.51763791709698032262542286379, 16.08970149343300944263190414352, 16.69949567191717073705808804127, 17.26445505774498131133380701502, 17.80626429427771496071314379051, 18.70921167357541409374851368369

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