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Properties

Label 1-3381-3381.1007-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: $\chi_{3381} (1007, \cdot )$
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

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

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