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Properties

Label 1-3381-3381.191-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.338 - 0.940i$
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  + (0.427 + 0.903i)2-s + (−0.634 + 0.773i)4-s + (0.760 − 0.649i)5-s + (−0.970 − 0.242i)8-s + (0.912 + 0.409i)10-s + (0.777 − 0.628i)11-s + (−0.714 + 0.699i)13-s + (−0.195 − 0.980i)16-s + (−0.352 + 0.935i)17-s + (−0.723 − 0.690i)19-s + (0.0203 + 0.999i)20-s + (0.900 + 0.433i)22-s + (0.155 − 0.987i)25-s + (−0.938 − 0.346i)26-s + (−0.986 + 0.162i)29-s + ⋯
L(s)  = 1  + (0.427 + 0.903i)2-s + (−0.634 + 0.773i)4-s + (0.760 − 0.649i)5-s + (−0.970 − 0.242i)8-s + (0.912 + 0.409i)10-s + (0.777 − 0.628i)11-s + (−0.714 + 0.699i)13-s + (−0.195 − 0.980i)16-s + (−0.352 + 0.935i)17-s + (−0.723 − 0.690i)19-s + (0.0203 + 0.999i)20-s + (0.900 + 0.433i)22-s + (0.155 − 0.987i)25-s + (−0.938 − 0.346i)26-s + (−0.986 + 0.162i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.338 - 0.940i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.338 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1171121853 - 0.1666329734i\)
\(L(\frac12)\) \(\approx\) \(0.1171121853 - 0.1666329734i\)
\(L(1)\) \(\approx\) \(0.9785710914 + 0.3874227396i\)
\(L(1)\) \(\approx\) \(0.9785710914 + 0.3874227396i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.427 + 0.903i)T \)
5 \( 1 + (0.760 - 0.649i)T \)
11 \( 1 + (0.777 - 0.628i)T \)
13 \( 1 + (-0.714 + 0.699i)T \)
17 \( 1 + (-0.352 + 0.935i)T \)
19 \( 1 + (-0.723 - 0.690i)T \)
29 \( 1 + (-0.986 + 0.162i)T \)
31 \( 1 + (-0.995 - 0.0950i)T \)
37 \( 1 + (0.115 + 0.993i)T \)
41 \( 1 + (-0.182 - 0.983i)T \)
43 \( 1 + (-0.970 + 0.242i)T \)
47 \( 1 + (-0.955 + 0.294i)T \)
53 \( 1 + (0.390 - 0.920i)T \)
59 \( 1 + (0.912 + 0.409i)T \)
61 \( 1 + (0.169 - 0.985i)T \)
67 \( 1 + (-0.928 - 0.371i)T \)
71 \( 1 + (0.301 - 0.953i)T \)
73 \( 1 + (-0.999 - 0.0135i)T \)
79 \( 1 + (-0.0475 + 0.998i)T \)
83 \( 1 + (-0.591 + 0.806i)T \)
89 \( 1 + (0.314 + 0.949i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
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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

−19.05113414859856262952469778951, −18.398294100086912373149166942716, −17.81089064115893772669284448364, −17.23516320524963148435577565901, −16.34557024962127740324776765575, −15.0919486717592640277546270759, −14.71434122774593191956002383972, −14.27857808903112914205966108599, −13.21434563149429986595524611966, −12.9483037301522056474491242008, −11.96247020751651954879751130991, −11.41691391120063296559151032330, −10.55912946669600432357793608647, −10.00400946506135090365148744639, −9.43291375202327301523260618807, −8.74762594398131927378862151513, −7.49236288724381097630989255636, −6.80067018583901742439865564223, −5.91331272354232255990374792943, −5.313045508822707893844878517600, −4.44260928237195222357494370007, −3.62408186478941356229453322607, −2.79677018214886456398990102671, −2.08556122103413188065601119405, −1.43152566202401369362206630864, 0.04608767118605197894513909511, 1.498528477146458279434452887902, 2.33826076789232613108808673997, 3.5341177618294832181109674626, 4.225669941027912928968174461831, 4.99308918002621299508912004366, 5.64789976320284991588313938111, 6.51521645285268608280505047344, 6.837930246909800005652507590749, 8.0115012251248489881815018160, 8.7001752409101912462965737526, 9.193021653559097168670852602130, 9.86926307373404007187653382547, 10.99720174900627591804918677584, 11.8295679899058562951323891002, 12.57728662256624632262660998347, 13.244803608592504570129293074725, 13.68013109487334752370954182348, 14.72416137554675442295882627852, 14.816693400157505937435917173723, 15.98774811546182250200939860510, 16.701106499326623990258417844399, 17.00709000443878994973293911297, 17.58972572233594815520005537181, 18.43540033071432121371283799948

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