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Properties

Label 1-3381-3381.1319-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.000137 + 0.999i$
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  + (0.0882 + 0.996i)2-s + (−0.984 + 0.175i)4-s + (0.440 + 0.897i)5-s + (−0.262 − 0.965i)8-s + (−0.855 + 0.517i)10-s + (−0.990 + 0.135i)11-s + (0.452 + 0.891i)13-s + (0.938 − 0.346i)16-s + (0.644 − 0.764i)17-s + (0.888 − 0.458i)19-s + (−0.591 − 0.806i)20-s + (−0.222 − 0.974i)22-s + (−0.612 + 0.790i)25-s + (−0.848 + 0.529i)26-s + (−0.339 − 0.940i)29-s + ⋯
L(s)  = 1  + (0.0882 + 0.996i)2-s + (−0.984 + 0.175i)4-s + (0.440 + 0.897i)5-s + (−0.262 − 0.965i)8-s + (−0.855 + 0.517i)10-s + (−0.990 + 0.135i)11-s + (0.452 + 0.891i)13-s + (0.938 − 0.346i)16-s + (0.644 − 0.764i)17-s + (0.888 − 0.458i)19-s + (−0.591 − 0.806i)20-s + (−0.222 − 0.974i)22-s + (−0.612 + 0.790i)25-s + (−0.848 + 0.529i)26-s + (−0.339 − 0.940i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000137 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000137 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.000137 + 0.999i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.000137 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.195587023 + 1.195422247i\)
\(L(\frac12)\) \(\approx\) \(1.195587023 + 1.195422247i\)
\(L(1)\) \(\approx\) \(0.9039345293 + 0.6321655979i\)
\(L(1)\) \(\approx\) \(0.9039345293 + 0.6321655979i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.0882 + 0.996i)T \)
5 \( 1 + (0.440 + 0.897i)T \)
11 \( 1 + (-0.990 + 0.135i)T \)
13 \( 1 + (0.452 + 0.891i)T \)
17 \( 1 + (0.644 - 0.764i)T \)
19 \( 1 + (0.888 - 0.458i)T \)
29 \( 1 + (-0.339 - 0.940i)T \)
31 \( 1 + (0.327 - 0.945i)T \)
37 \( 1 + (-0.568 + 0.822i)T \)
41 \( 1 + (0.557 - 0.830i)T \)
43 \( 1 + (0.262 - 0.965i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (0.923 + 0.384i)T \)
59 \( 1 + (0.855 - 0.517i)T \)
61 \( 1 + (0.0339 - 0.999i)T \)
67 \( 1 + (0.235 + 0.971i)T \)
71 \( 1 + (0.0611 - 0.998i)T \)
73 \( 1 + (-0.810 + 0.585i)T \)
79 \( 1 + (0.580 + 0.814i)T \)
83 \( 1 + (0.685 - 0.728i)T \)
89 \( 1 + (0.534 - 0.844i)T \)
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

−18.64469799614951846203321036892, −17.89264261687734660355093419185, −17.68304597943880069457117168784, −16.5083722708792797597875653067, −16.11038710343750922383416354006, −15.03840725102503758864419811325, −14.30217188821825031426773143973, −13.46248854203717858240764083611, −13.03410508615439768520093596753, −12.41834538836834677864070754272, −11.82751999545588825705186841776, −10.70316006149414364694799256421, −10.37597816597906011775312507704, −9.63185267031219095549031933904, −8.7979647908274150774309707848, −8.22877676493450144107541920219, −7.55641714693272446651526576572, −6.06219864420711067020820122097, −5.3870756985896992446504131635, −5.05457172759276976489268495155, −3.910187501844725335039995421730, −3.25047806230571510957073601571, −2.38553883424151280988275639494, −1.410511393127452707358035523131, −0.80517299452390874423561116742, 0.695714622919414668275349942310, 2.09427623176595428537886706345, 2.95094504021521667927408047074, 3.7698564969446242034915201037, 4.694198812456126506989550768413, 5.53997555938212382431376299190, 6.02530209626289005267690663029, 7.0077863431956948052726882229, 7.37752691301634966713338034353, 8.14507863953318369595002505322, 9.14942487998947355390273112875, 9.71617402809791363528579005933, 10.35858652904979905379735389235, 11.323277776059687976573334472671, 12.02170995448717390887557762419, 13.10241350358615229522029942907, 13.71608449762977017805724190288, 14.06041211957911545914774584340, 14.885590887433017063753905068260, 15.68102518049532709680256048546, 15.96878312417782863813269042340, 17.02130057906321613034127962753, 17.49239475962375089200688645041, 18.41768785695287559960221333535, 18.62790235787829286925615222756

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