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Properties

Label 1-3381-3381.401-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.770 - 0.637i$
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  + (0.997 + 0.0679i)2-s + (0.990 + 0.135i)4-s + (0.994 + 0.108i)5-s + (0.979 + 0.202i)8-s + (0.984 + 0.175i)10-s + (−0.675 − 0.737i)11-s + (−0.992 + 0.122i)13-s + (0.963 + 0.268i)16-s + (−0.612 − 0.790i)17-s + (−0.235 − 0.971i)19-s + (0.970 + 0.242i)20-s + (−0.623 − 0.781i)22-s + (0.976 + 0.215i)25-s + (−0.998 + 0.0543i)26-s + (0.377 − 0.925i)29-s + ⋯
L(s)  = 1  + (0.997 + 0.0679i)2-s + (0.990 + 0.135i)4-s + (0.994 + 0.108i)5-s + (0.979 + 0.202i)8-s + (0.984 + 0.175i)10-s + (−0.675 − 0.737i)11-s + (−0.992 + 0.122i)13-s + (0.963 + 0.268i)16-s + (−0.612 − 0.790i)17-s + (−0.235 − 0.971i)19-s + (0.970 + 0.242i)20-s + (−0.623 − 0.781i)22-s + (0.976 + 0.215i)25-s + (−0.998 + 0.0543i)26-s + (0.377 − 0.925i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.660853281 - 1.317555875i\)
\(L(\frac12)\) \(\approx\) \(3.660853281 - 1.317555875i\)
\(L(1)\) \(\approx\) \(2.194840380 - 0.1904397530i\)
\(L(1)\) \(\approx\) \(2.194840380 - 0.1904397530i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.997 + 0.0679i)T \)
5 \( 1 + (0.994 + 0.108i)T \)
11 \( 1 + (-0.675 - 0.737i)T \)
13 \( 1 + (-0.992 + 0.122i)T \)
17 \( 1 + (-0.612 - 0.790i)T \)
19 \( 1 + (-0.235 - 0.971i)T \)
29 \( 1 + (0.377 - 0.925i)T \)
31 \( 1 + (0.580 + 0.814i)T \)
37 \( 1 + (-0.760 + 0.649i)T \)
41 \( 1 + (0.591 - 0.806i)T \)
43 \( 1 + (0.979 - 0.202i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (-0.912 - 0.409i)T \)
59 \( 1 + (0.984 + 0.175i)T \)
61 \( 1 + (0.546 - 0.837i)T \)
67 \( 1 + (0.786 + 0.618i)T \)
71 \( 1 + (0.862 - 0.505i)T \)
73 \( 1 + (-0.352 + 0.935i)T \)
79 \( 1 + (0.888 + 0.458i)T \)
83 \( 1 + (0.262 - 0.965i)T \)
89 \( 1 + (-0.169 + 0.985i)T \)
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

−19.08791897405678243361066788442, −18.056631265146880697626636586059, −17.4223098354824714913760076042, −16.833644552109955128677574334394, −15.993643321409697186711902418893, −15.29773832043755524736169596744, −14.48687639667708709484788305953, −14.22137420307569786397491276857, −13.17474932966289568231785162408, −12.69755226699607980926036259551, −12.31537426465923759803721575302, −11.20809679500460413969947769040, −10.42017362037040959581219134598, −10.03491549267621980906565349679, −9.16511115678374075648150727672, −8.02614335923969848714858382132, −7.3775251325292328546503199799, −6.459042800759088730438430173937, −5.93769679854823142461122447678, −5.07033506526872976224222768667, −4.601450448265934361090743618657, −3.644901995742706799225544099908, −2.512119347469463291534680017005, −2.198098770653673398414200193644, −1.248090810227522989872242065, 0.74143580648555161092564336464, 2.14925921517021508535625290460, 2.50033100787532606610581490610, 3.26460108351514817304222680465, 4.43439291598816348775962191408, 5.11688752912716429808249779158, 5.56812087591354346215093685562, 6.64145230200103486159277923047, 6.91046789494928214002424227692, 7.962179782984589527590203531001, 8.83722148796735027871284950733, 9.749357747063279513949616583347, 10.43692826141372590384832219595, 11.12260391760761459246741085201, 11.833865327963418440198773600153, 12.696229511372434203310712586108, 13.27943334356067548284573385884, 13.94905904715081176403167438414, 14.26187746386722718375476104210, 15.34707040263996889465408528914, 15.759540365632523410353586317394, 16.59269876641141755233557203702, 17.3904831040446294090894814333, 17.77640064696353374100489646437, 18.95216018827390385865579049378

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