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Properties

Label 1-3381-3381.1343-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.591 - 0.806i$
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  + (−0.947 − 0.320i)2-s + (0.794 + 0.607i)4-s + (0.742 − 0.670i)5-s + (−0.557 − 0.830i)8-s + (−0.917 + 0.396i)10-s + (−0.101 − 0.994i)11-s + (0.999 + 0.0407i)13-s + (0.262 + 0.965i)16-s + (0.794 − 0.607i)17-s + (−0.415 + 0.909i)19-s + (0.996 − 0.0815i)20-s + (−0.222 + 0.974i)22-s + (0.101 − 0.994i)25-s + (−0.933 − 0.359i)26-s + (−0.794 + 0.607i)29-s + ⋯
L(s)  = 1  + (−0.947 − 0.320i)2-s + (0.794 + 0.607i)4-s + (0.742 − 0.670i)5-s + (−0.557 − 0.830i)8-s + (−0.917 + 0.396i)10-s + (−0.101 − 0.994i)11-s + (0.999 + 0.0407i)13-s + (0.262 + 0.965i)16-s + (0.794 − 0.607i)17-s + (−0.415 + 0.909i)19-s + (0.996 − 0.0815i)20-s + (−0.222 + 0.974i)22-s + (0.101 − 0.994i)25-s + (−0.933 − 0.359i)26-s + (−0.794 + 0.607i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.591 - 0.806i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.591 - 0.806i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.284180942 - 0.6508299504i\)
\(L(\frac12)\) \(\approx\) \(1.284180942 - 0.6508299504i\)
\(L(1)\) \(\approx\) \(0.8706328320 - 0.2492854807i\)
\(L(1)\) \(\approx\) \(0.8706328320 - 0.2492854807i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.947 - 0.320i)T \)
5 \( 1 + (0.742 - 0.670i)T \)
11 \( 1 + (-0.101 - 0.994i)T \)
13 \( 1 + (0.999 + 0.0407i)T \)
17 \( 1 + (0.794 - 0.607i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (-0.794 + 0.607i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (-0.0611 + 0.998i)T \)
41 \( 1 + (0.742 - 0.670i)T \)
43 \( 1 + (0.557 - 0.830i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (-0.882 + 0.470i)T \)
59 \( 1 + (0.917 - 0.396i)T \)
61 \( 1 + (0.933 - 0.359i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.339 - 0.940i)T \)
73 \( 1 + (0.452 + 0.891i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (-0.818 + 0.574i)T \)
89 \( 1 + (0.685 + 0.728i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
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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

−18.78425058041427199309228259670, −18.182256285085527073416049169654, −17.4893812349437661778494565917, −17.18402967583081845490678561716, −16.20213660718089688365906285671, −15.53450752110120759550720274937, −14.76503447973179055204163967887, −14.45447572307976635797424738489, −13.31717582773414502840689896300, −12.79164795996032806695529088041, −11.57947819962186434918622869642, −11.07125718096780400927710241522, −10.325072169032432869417135705693, −9.75847165092528689152121435284, −9.151464911376154653066207673202, −8.29765069590839295778160405954, −7.46852066564177554939823244731, −6.936921212370523337544182015226, −5.98371521186220148601882512739, −5.73058162514421991131649861551, −4.49221523418496052749137119990, −3.41916984296428437926774391119, −2.37675059517561900441937806113, −1.900340038133823448312023717968, −0.85213059640057713000733953776, 0.79370570008274319897161833843, 1.368047462148927080681033070172, 2.27397976099761755557862240451, 3.25055204866774891834246089622, 3.88731859824225635423586496150, 5.19822592799678857555169865863, 5.89722438719827261052245313617, 6.50993651556080809095328313509, 7.55133743299719711682702269625, 8.34273603114349886635044930024, 8.8019250956012570114509340914, 9.479376837485820758937305816970, 10.26621885619395560813489472765, 10.8452053008881929239868047531, 11.59205266550020019997541440074, 12.45527612308284416499309440019, 12.92890664497807169511497087801, 13.89124811448792858928610545215, 14.36994762024242263423891195828, 15.8186116643148703873910036591, 15.99012533410830904817940206099, 16.88602106112070794665596159379, 17.183452021356047066377548941635, 18.219116058249757587719835937354, 18.61942264870881690742315913311

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