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Properties

Label 1-3381-3381.1112-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.664 - 0.747i$
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  + (−0.742 − 0.670i)2-s + (0.101 + 0.994i)4-s + (0.996 − 0.0815i)5-s + (0.591 − 0.806i)8-s + (−0.794 − 0.607i)10-s + (−0.986 − 0.162i)11-s + (0.768 − 0.639i)13-s + (−0.979 + 0.202i)16-s + (0.101 − 0.994i)17-s + (−0.841 − 0.540i)19-s + (0.182 + 0.983i)20-s + (0.623 + 0.781i)22-s + (0.986 − 0.162i)25-s + (−0.999 − 0.0407i)26-s + (−0.101 + 0.994i)29-s + ⋯
L(s)  = 1  + (−0.742 − 0.670i)2-s + (0.101 + 0.994i)4-s + (0.996 − 0.0815i)5-s + (0.591 − 0.806i)8-s + (−0.794 − 0.607i)10-s + (−0.986 − 0.162i)11-s + (0.768 − 0.639i)13-s + (−0.979 + 0.202i)16-s + (0.101 − 0.994i)17-s + (−0.841 − 0.540i)19-s + (0.182 + 0.983i)20-s + (0.623 + 0.781i)22-s + (0.986 − 0.162i)25-s + (−0.999 − 0.0407i)26-s + (−0.101 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.664 - 0.747i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.664 - 0.747i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4434109148 - 0.9875685393i\)
\(L(\frac12)\) \(\approx\) \(0.4434109148 - 0.9875685393i\)
\(L(1)\) \(\approx\) \(0.7534924289 - 0.3554181505i\)
\(L(1)\) \(\approx\) \(0.7534924289 - 0.3554181505i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.742 - 0.670i)T \)
5 \( 1 + (0.996 - 0.0815i)T \)
11 \( 1 + (-0.986 - 0.162i)T \)
13 \( 1 + (0.768 - 0.639i)T \)
17 \( 1 + (0.101 - 0.994i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (-0.101 + 0.994i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + (-0.862 - 0.505i)T \)
41 \( 1 + (0.996 - 0.0815i)T \)
43 \( 1 + (-0.591 - 0.806i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.452 - 0.891i)T \)
59 \( 1 + (0.794 + 0.607i)T \)
61 \( 1 + (0.999 - 0.0407i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (0.377 + 0.925i)T \)
73 \( 1 + (0.992 + 0.122i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.557 + 0.830i)T \)
89 \( 1 + (0.262 - 0.965i)T \)
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

−18.946898537254379313162994187025, −18.28637797977439456175640287455, −17.63832953469144687922825423107, −17.07535182052859582284864269659, −16.4434116371904087898663890302, −15.72022241145994954127125835787, −14.947013785969277831265356918071, −14.43680069155616462169893570035, −13.43830453022505188889034895252, −13.21657180996704931646699658486, −12.05661223947822681613392018772, −10.951294999821284394074274629893, −10.5561346993459252677521326363, −9.81108234574942345193736821580, −9.25819778316608570345586262471, −8.22729267867771364719625544181, −8.01850889925605660624082229085, −6.75479248054682358182027827855, −6.26494697302634631567187112731, −5.69472899613108406866065291334, −4.847489151115276189382080744423, −3.93469670017850842857765339480, −2.55373443369463168409889469760, −1.9295372951723173933283520754, −1.116501596965379366448187365345, 0.42868817967510106339623312797, 1.37977560557308726930218655856, 2.313887149063294496642860831774, 2.87237815265839931415560895285, 3.711310740244440958446940724358, 4.93127069703435578687588425212, 5.46275129196627121427367703205, 6.59218077110446893641673410423, 7.16551779728845959738668952810, 8.24802527259730338682758450768, 8.69165684398487742569236160149, 9.435212587590111736298230008829, 10.30297080341519654429312499254, 10.60310068646353792923139025339, 11.35838868974867170567343746063, 12.327068555306538560851446232833, 13.0271178375623258525573393133, 13.42266089442902271240107361291, 14.19435051642105105583664462289, 15.28702830859588549208437138387, 16.10433445829024705287289202727, 16.505942300243580877324359300346, 17.60961160679530746941174195842, 17.83676655547031815262903593667, 18.44561512238806418750374300693

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