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Properties

Label 1-3381-3381.527-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.389 - 0.921i$
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  + (−0.998 + 0.0543i)2-s + (0.994 − 0.108i)4-s + (0.390 + 0.920i)5-s + (−0.986 + 0.162i)8-s + (−0.440 − 0.897i)10-s + (−0.275 + 0.961i)11-s + (−0.862 + 0.505i)13-s + (0.976 − 0.215i)16-s + (−0.403 − 0.915i)17-s + (−0.981 − 0.189i)19-s + (0.488 + 0.872i)20-s + (0.222 − 0.974i)22-s + (−0.694 + 0.719i)25-s + (0.833 − 0.552i)26-s + (0.591 − 0.806i)29-s + ⋯
L(s)  = 1  + (−0.998 + 0.0543i)2-s + (0.994 − 0.108i)4-s + (0.390 + 0.920i)5-s + (−0.986 + 0.162i)8-s + (−0.440 − 0.897i)10-s + (−0.275 + 0.961i)11-s + (−0.862 + 0.505i)13-s + (0.976 − 0.215i)16-s + (−0.403 − 0.915i)17-s + (−0.981 − 0.189i)19-s + (0.488 + 0.872i)20-s + (0.222 − 0.974i)22-s + (−0.694 + 0.719i)25-s + (0.833 − 0.552i)26-s + (0.591 − 0.806i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.389 - 0.921i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.389 - 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3786631896 - 0.2510828700i\)
\(L(\frac12)\) \(\approx\) \(0.3786631896 - 0.2510828700i\)
\(L(1)\) \(\approx\) \(0.5998387771 + 0.09915592533i\)
\(L(1)\) \(\approx\) \(0.5998387771 + 0.09915592533i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.998 + 0.0543i)T \)
5 \( 1 + (0.390 + 0.920i)T \)
11 \( 1 + (-0.275 + 0.961i)T \)
13 \( 1 + (-0.862 + 0.505i)T \)
17 \( 1 + (-0.403 - 0.915i)T \)
19 \( 1 + (-0.981 - 0.189i)T \)
29 \( 1 + (0.591 - 0.806i)T \)
31 \( 1 + (0.723 - 0.690i)T \)
37 \( 1 + (0.248 - 0.968i)T \)
41 \( 1 + (0.992 + 0.122i)T \)
43 \( 1 + (-0.986 - 0.162i)T \)
47 \( 1 + (-0.365 + 0.930i)T \)
53 \( 1 + (-0.568 + 0.822i)T \)
59 \( 1 + (-0.440 - 0.897i)T \)
61 \( 1 + (-0.894 + 0.446i)T \)
67 \( 1 + (0.995 + 0.0950i)T \)
71 \( 1 + (0.979 - 0.202i)T \)
73 \( 1 + (-0.942 - 0.333i)T \)
79 \( 1 + (-0.928 + 0.371i)T \)
83 \( 1 + (0.101 - 0.994i)T \)
89 \( 1 + (0.990 - 0.135i)T \)
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

−19.03080262359972863799848661584, −18.137976876326569614824653220779, −17.50915549276386114986493542069, −16.92544692445318672599301169972, −16.464350399533658851242597308504, −15.67418480248320038671946745528, −15.020572226909012293203535794666, −14.15680049216879966203978810432, −13.157543845399239254469871388852, −12.61477143460998784344482096583, −11.96721547140488446587638317149, −11.0516168989963006574612216047, −10.35966547899440589855511530490, −9.84907040230705808070372671758, −8.90336119591843874100281017912, −8.36432235592763431443681375313, −7.98514635900394024662133172919, −6.78168374755189688869224110421, −6.18351053825199897539565004562, −5.3822607560937174832214180606, −4.557572601657528520123500837119, −3.4062303928796969515042495846, −2.56322925386181995202101564332, −1.71251938413068212387685360315, −0.880081836253651637930462406987, 0.210783069126847328465929728961, 1.68252729233717777248050679015, 2.44897917329538973798453414572, 2.76932256670923660680206045742, 4.16663969640704063275124727821, 5.000608298655725062350748009934, 6.17949152449330623285386833861, 6.59830532511951922033722509070, 7.42832289275866695863249970520, 7.817770313963288473364287285701, 9.00145924390994905407709448003, 9.61680667088319826518844328887, 10.078215946926133071649381852590, 10.86377780503115009641608515706, 11.48789948454265546245471093838, 12.21742150194222424904444568890, 13.04951285618537606001114490117, 14.0851178466221107748900643818, 14.66867909762288276197395025898, 15.35509388655285535449057542034, 15.87226421242438205855862447610, 16.92694220519074545169045081564, 17.43213359880689866855503366206, 17.92896871187485700189894570667, 18.67650550237280653115542242922

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