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Properties

Label 1-3381-3381.548-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.763 - 0.646i$
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  + (0.802 + 0.596i)2-s + (0.288 + 0.957i)4-s + (−0.923 − 0.384i)5-s + (−0.339 + 0.940i)8-s + (−0.511 − 0.859i)10-s + (−0.966 − 0.255i)11-s + (0.742 + 0.670i)13-s + (−0.833 + 0.552i)16-s + (−0.973 + 0.229i)17-s + (0.786 − 0.618i)19-s + (0.101 − 0.994i)20-s + (−0.623 − 0.781i)22-s + (0.704 + 0.709i)25-s + (0.195 + 0.980i)26-s + (−0.685 − 0.728i)29-s + ⋯
L(s)  = 1  + (0.802 + 0.596i)2-s + (0.288 + 0.957i)4-s + (−0.923 − 0.384i)5-s + (−0.339 + 0.940i)8-s + (−0.511 − 0.859i)10-s + (−0.966 − 0.255i)11-s + (0.742 + 0.670i)13-s + (−0.833 + 0.552i)16-s + (−0.973 + 0.229i)17-s + (0.786 − 0.618i)19-s + (0.101 − 0.994i)20-s + (−0.623 − 0.781i)22-s + (0.704 + 0.709i)25-s + (0.195 + 0.980i)26-s + (−0.685 − 0.728i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.763 - 0.646i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (548, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.763 - 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066011514 - 0.3907310070i\)
\(L(\frac12)\) \(\approx\) \(1.066011514 - 0.3907310070i\)
\(L(1)\) \(\approx\) \(1.106597859 + 0.3268573568i\)
\(L(1)\) \(\approx\) \(1.106597859 + 0.3268573568i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.802 + 0.596i)T \)
5 \( 1 + (-0.923 - 0.384i)T \)
11 \( 1 + (-0.966 - 0.255i)T \)
13 \( 1 + (0.742 + 0.670i)T \)
17 \( 1 + (-0.973 + 0.229i)T \)
19 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (-0.685 - 0.728i)T \)
31 \( 1 + (-0.888 + 0.458i)T \)
37 \( 1 + (0.546 - 0.837i)T \)
41 \( 1 + (-0.794 + 0.607i)T \)
43 \( 1 + (-0.339 - 0.940i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (0.906 - 0.421i)T \)
59 \( 1 + (-0.511 - 0.859i)T \)
61 \( 1 + (0.751 + 0.659i)T \)
67 \( 1 + (0.327 + 0.945i)T \)
71 \( 1 + (0.999 + 0.0407i)T \)
73 \( 1 + (-0.997 + 0.0679i)T \)
79 \( 1 + (-0.235 - 0.971i)T \)
83 \( 1 + (0.0203 + 0.999i)T \)
89 \( 1 + (0.999 + 0.0271i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−18.80146454936547141099043718716, −18.458376397143454035709369156212, −17.885959447436683584746643431859, −16.49157161206477314879263854461, −15.89963673658112961598277091452, −15.31076450066635598518989911863, −14.841607855926498817611017510073, −13.94108032421590204096566296622, −13.20476836706910784849120290514, −12.708657298178750480390699953392, −11.89883377071966263822506656595, −11.174080978923797854764233897380, −10.76696449052937944363806273515, −10.031075226748144736830193319123, −9.12372548979851907750227613704, −8.12433073312660856092593879704, −7.45439489445861643627967053977, −6.67353317836306651640947500469, −5.78256578489655697785727382380, −5.0745909320295226032207487262, −4.29245450330458387706795915135, −3.47849472501078218437829439877, −2.96380702383332386176741582616, −2.04478398146251595405183714180, −0.958222092577220321318122991068, 0.2779691929476364298722391606, 1.822536698631129636912813312677, 2.77590742869451237962048313914, 3.70312900568331473513339604560, 4.15470283465438042988566294845, 5.10305921521230308904376505370, 5.581279646919107582271290394740, 6.6947682502340456694131616427, 7.20153937822575009613309907763, 8.02489342766914521693144879576, 8.60654145213973520520708674894, 9.25565013682296756889562872250, 10.604344711941283053910435024542, 11.37718045050683985218710179971, 11.67955099441440639139148514401, 12.69827563331545766379833000940, 13.27718402519699098521314021796, 13.70563429877990453959940960826, 14.78189262285213741825407606255, 15.32170497824044695561355516996, 16.01274890335291132023623043771, 16.29674639235338582629742417032, 17.1447688794391738154385751370, 18.048523305204090830192491954913, 18.60725406952518615717223603814

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