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Properties

Label 1-3381-3381.1184-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.950 + 0.310i$
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  + (−0.794 − 0.607i)2-s + (0.262 + 0.965i)4-s + (0.101 − 0.994i)5-s + (0.377 − 0.925i)8-s + (−0.685 + 0.728i)10-s + (−0.979 + 0.202i)11-s + (0.996 + 0.0815i)13-s + (−0.862 + 0.505i)16-s + (0.262 − 0.965i)17-s + (0.654 + 0.755i)19-s + (0.986 − 0.162i)20-s + (0.900 + 0.433i)22-s + (−0.979 − 0.202i)25-s + (−0.742 − 0.670i)26-s + (−0.262 + 0.965i)29-s + ⋯
L(s)  = 1  + (−0.794 − 0.607i)2-s + (0.262 + 0.965i)4-s + (0.101 − 0.994i)5-s + (0.377 − 0.925i)8-s + (−0.685 + 0.728i)10-s + (−0.979 + 0.202i)11-s + (0.996 + 0.0815i)13-s + (−0.862 + 0.505i)16-s + (0.262 − 0.965i)17-s + (0.654 + 0.755i)19-s + (0.986 − 0.162i)20-s + (0.900 + 0.433i)22-s + (−0.979 − 0.202i)25-s + (−0.742 − 0.670i)26-s + (−0.262 + 0.965i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.950 + 0.310i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.950 + 0.310i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8365706446 + 0.1333627498i\)
\(L(\frac12)\) \(\approx\) \(0.8365706446 + 0.1333627498i\)
\(L(1)\) \(\approx\) \(0.6903507664 - 0.1903604982i\)
\(L(1)\) \(\approx\) \(0.6903507664 - 0.1903604982i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.794 - 0.607i)T \)
5 \( 1 + (0.101 - 0.994i)T \)
11 \( 1 + (-0.979 + 0.202i)T \)
13 \( 1 + (0.996 + 0.0815i)T \)
17 \( 1 + (0.262 - 0.965i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.262 + 0.965i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + (0.992 + 0.122i)T \)
41 \( 1 + (-0.101 + 0.994i)T \)
43 \( 1 + (0.377 + 0.925i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (0.557 - 0.830i)T \)
59 \( 1 + (-0.685 + 0.728i)T \)
61 \( 1 + (-0.742 + 0.670i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
71 \( 1 + (0.768 - 0.639i)T \)
73 \( 1 + (-0.591 + 0.806i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (0.339 - 0.940i)T \)
89 \( 1 + (-0.0611 + 0.998i)T \)
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

−18.72705178175921597358470608811, −18.10131786115862721527691812086, −17.54998501094138569066102712348, −16.773725225711635701041617583630, −15.93115031019141654970140801248, −15.37876396034785627025652613189, −14.954473130889655620855738048531, −13.8922748251599388725680085379, −13.60958357559937954708009146835, −12.5523448716561641622019730805, −11.35060346146952169644553531981, −10.89548423956794854530621403280, −10.4205199526899490210465352476, −9.56486364784804447100299520599, −8.88776112802785949710319477069, −7.887595353112483915556771648559, −7.59715754017532845621069260149, −6.67797532904486854200557505139, −5.89530151116242998144748794238, −5.55120092114833963499626850487, −4.29522578737386809541544322336, −3.29357555052472722895138084487, −2.44498828180420383576792876930, −1.61873662301964854520688748511, −0.38502757923445015937830522474, 0.95133088599845931222752957246, 1.52052378266569883253781643168, 2.55475523611358347822873874102, 3.370552292333433997137589381362, 4.21058775062233470019401499504, 5.1235963469189773525270699026, 5.8201009037184495914807186468, 6.972156341260362363182400026864, 7.770436748602053402312783908866, 8.26524334989704561935513056992, 9.09300323388295447091832554206, 9.59891681981720954920334260438, 10.366648840737871937233644557649, 11.12093250413534485865722547945, 11.82943225228496825596194508472, 12.491772700436358370550347528547, 13.20112739809631145728163258078, 13.61110705437845460627326584499, 14.76863152554098736969271344565, 15.82778802261706125765518529235, 16.38417105395259541277173765089, 16.54923984085820580276411302912, 17.72596122546043657862020399057, 18.345923980362142253743951130794, 18.513234512378907070586253082112

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