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Properties

Label 1-3381-3381.1139-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.990 - 0.138i$
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  + (0.694 + 0.719i)2-s + (−0.0339 + 0.999i)4-s + (0.999 + 0.0271i)5-s + (−0.742 + 0.670i)8-s + (0.675 + 0.737i)10-s + (0.546 + 0.837i)11-s + (−0.685 + 0.728i)13-s + (−0.997 − 0.0679i)16-s + (−0.848 + 0.529i)17-s + (0.327 − 0.945i)19-s + (−0.0611 + 0.998i)20-s + (−0.222 + 0.974i)22-s + (0.998 + 0.0543i)25-s + (−0.999 + 0.0135i)26-s + (−0.882 − 0.470i)29-s + ⋯
L(s)  = 1  + (0.694 + 0.719i)2-s + (−0.0339 + 0.999i)4-s + (0.999 + 0.0271i)5-s + (−0.742 + 0.670i)8-s + (0.675 + 0.737i)10-s + (0.546 + 0.837i)11-s + (−0.685 + 0.728i)13-s + (−0.997 − 0.0679i)16-s + (−0.848 + 0.529i)17-s + (0.327 − 0.945i)19-s + (−0.0611 + 0.998i)20-s + (−0.222 + 0.974i)22-s + (0.998 + 0.0543i)25-s + (−0.999 + 0.0135i)26-s + (−0.882 − 0.470i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.990 - 0.138i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.990 - 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1426506731 + 2.045830962i\)
\(L(\frac12)\) \(\approx\) \(-0.1426506731 + 2.045830962i\)
\(L(1)\) \(\approx\) \(1.156365518 + 0.9867582709i\)
\(L(1)\) \(\approx\) \(1.156365518 + 0.9867582709i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.694 + 0.719i)T \)
5 \( 1 + (0.999 + 0.0271i)T \)
11 \( 1 + (0.546 + 0.837i)T \)
13 \( 1 + (-0.685 + 0.728i)T \)
17 \( 1 + (-0.848 + 0.529i)T \)
19 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (-0.882 - 0.470i)T \)
31 \( 1 + (-0.235 + 0.971i)T \)
37 \( 1 + (-0.984 + 0.175i)T \)
41 \( 1 + (-0.523 + 0.852i)T \)
43 \( 1 + (0.742 + 0.670i)T \)
47 \( 1 + (0.365 - 0.930i)T \)
53 \( 1 + (-0.777 + 0.628i)T \)
59 \( 1 + (-0.675 - 0.737i)T \)
61 \( 1 + (-0.511 + 0.859i)T \)
67 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (-0.794 - 0.607i)T \)
73 \( 1 + (-0.464 + 0.885i)T \)
79 \( 1 + (-0.786 + 0.618i)T \)
83 \( 1 + (0.947 - 0.320i)T \)
89 \( 1 + (0.906 + 0.421i)T \)
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

−18.71569708307260173397058293333, −17.80382253723738864814565113667, −17.20903756699310627460618302984, −16.3688169438292979001055478032, −15.53823357361791254090372209995, −14.73094988534415069293624536320, −14.10222408883918786826503325541, −13.64882379040302529784341674023, −12.88723767806811945775198496723, −12.29510755724261075320989065122, −11.48881993883791087906608778595, −10.71711392842585925958439194803, −10.2246110523157609954936102788, −9.26690704798526168620713353268, −9.01261510564963758542428830038, −7.70425407408596448676350880270, −6.75734446313626529016803989513, −5.9552236603719440558939999450, −5.49056226371448473326562719740, −4.73876716075931470249947730964, −3.729176759037759339692609466872, −3.04824418727356665163037926040, −2.18240199277112790578010108102, −1.52313816744012090893282917273, −0.40690047477791388539288211517, 1.593770450050459157279621541776, 2.26899224026620834475655141727, 3.12082412991409785064721896912, 4.23565303732729915614442223624, 4.75456566900347227752546996297, 5.49255163002379800349275815298, 6.40144423630854642121975932788, 6.87335332520187646896065732346, 7.47534490208373040979718061465, 8.67362432295449313216484228882, 9.182515918125601160765776433314, 9.84017163409430113689798825239, 10.877074760422578306964460128931, 11.687895225966604626194727140332, 12.42904843135697805341509104893, 13.070604921729523208516880951325, 13.71417888663449395066658560928, 14.3427350801880802124971809727, 14.950281904614153901859502696052, 15.57252790287564715713203789244, 16.48639528506994525661137502996, 17.16510911284971753197266802941, 17.54188557449196584980089812596, 18.14165751009444658674484177169, 19.17443289328476782925600878013

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