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Properties

Label 1-3381-3381.743-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.338 - 0.940i$
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  + (−0.996 − 0.0815i)2-s + (0.986 + 0.162i)4-s + (0.182 + 0.983i)5-s + (−0.970 − 0.242i)8-s + (−0.101 − 0.994i)10-s + (−0.933 − 0.359i)11-s + (−0.714 + 0.699i)13-s + (0.947 + 0.320i)16-s + (0.986 − 0.162i)17-s + (0.959 − 0.281i)19-s + (0.0203 + 0.999i)20-s + (0.900 + 0.433i)22-s + (−0.933 + 0.359i)25-s + (0.768 − 0.639i)26-s + (−0.986 + 0.162i)29-s + ⋯
L(s)  = 1  + (−0.996 − 0.0815i)2-s + (0.986 + 0.162i)4-s + (0.182 + 0.983i)5-s + (−0.970 − 0.242i)8-s + (−0.101 − 0.994i)10-s + (−0.933 − 0.359i)11-s + (−0.714 + 0.699i)13-s + (0.947 + 0.320i)16-s + (0.986 − 0.162i)17-s + (0.959 − 0.281i)19-s + (0.0203 + 0.999i)20-s + (0.900 + 0.433i)22-s + (−0.933 + 0.359i)25-s + (0.768 − 0.639i)26-s + (−0.986 + 0.162i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.338 - 0.940i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.338 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1404257914 - 0.1998047182i\)
\(L(\frac12)\) \(\approx\) \(0.1404257914 - 0.1998047182i\)
\(L(1)\) \(\approx\) \(0.5810498829 + 0.07030268395i\)
\(L(1)\) \(\approx\) \(0.5810498829 + 0.07030268395i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.996 - 0.0815i)T \)
5 \( 1 + (0.182 + 0.983i)T \)
11 \( 1 + (-0.933 - 0.359i)T \)
13 \( 1 + (-0.714 + 0.699i)T \)
17 \( 1 + (0.986 - 0.162i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (-0.986 + 0.162i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (-0.917 - 0.396i)T \)
41 \( 1 + (-0.182 - 0.983i)T \)
43 \( 1 + (-0.970 + 0.242i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (-0.992 + 0.122i)T \)
59 \( 1 + (-0.101 - 0.994i)T \)
61 \( 1 + (0.768 + 0.639i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (0.301 - 0.953i)T \)
73 \( 1 + (0.488 + 0.872i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (-0.591 + 0.806i)T \)
89 \( 1 + (-0.979 - 0.202i)T \)
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

−18.84136973491575290075330150194, −18.39302758534186292925187639829, −17.51036907108584340400597209125, −17.0685166136719668216398053386, −16.42577146003495439437111251929, −15.71662585410419891316687071655, −15.17101628806880814354539391282, −14.32492829856030241435979745774, −13.30636564341839909065284292569, −12.61792812399054640064285850981, −12.0431165513291409992393924601, −11.30275826117628662931799801649, −10.24598654316097282065980864659, −9.85168866459126886414442520308, −9.29976900981295469388496433161, −8.20085742902301817618012714790, −7.89176489173667993267707262635, −7.24390623470659748601080791091, −6.06513587150339428877446054965, −5.410763205983608819125980201458, −4.85955768230204032252334678460, −3.51367339852891321773019972014, −2.68570787703600710299133428356, −1.77314731027203271353725281567, −0.97839618493231067298599991919, 0.109350243680792338635072107533, 1.47462474961109000917334027963, 2.293536566563426387838788034969, 3.05078110561196042891771634703, 3.6232066005265250874428166654, 5.16869747410667156053885068890, 5.70932515968795025200476647928, 6.83176122745866737439384734294, 7.19282297311082454271177664720, 7.88558319812339277128220927159, 8.72497076085605800995575367787, 9.63859264602011851170118590029, 10.08022349200494291966218596955, 10.74513087085912616113283084810, 11.48565542610369728108840228534, 12.02830146190624594031336550459, 12.94258106748063791018318047709, 14.004719686878995960701347353335, 14.4292587374781164928577240893, 15.39170953984109091428978346022, 15.87248059961527036385174840299, 16.67296138287128660941642377166, 17.32167176155419188630379009830, 18.08479670494774637404831931931, 18.614761043065300772309860276672

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