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Properties

Label 1-3381-3381.11-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.976 + 0.216i$
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  + (0.876 + 0.482i)2-s + (0.534 + 0.844i)4-s + (−0.984 − 0.175i)5-s + (0.0611 + 0.998i)8-s + (−0.777 − 0.628i)10-s + (−0.169 + 0.985i)11-s + (0.557 + 0.830i)13-s + (−0.427 + 0.903i)16-s + (0.464 + 0.885i)17-s + (−0.981 + 0.189i)19-s + (−0.377 − 0.925i)20-s + (−0.623 + 0.781i)22-s + (0.938 + 0.346i)25-s + (0.0882 + 0.996i)26-s + (0.999 + 0.0407i)29-s + ⋯
L(s)  = 1  + (0.876 + 0.482i)2-s + (0.534 + 0.844i)4-s + (−0.984 − 0.175i)5-s + (0.0611 + 0.998i)8-s + (−0.777 − 0.628i)10-s + (−0.169 + 0.985i)11-s + (0.557 + 0.830i)13-s + (−0.427 + 0.903i)16-s + (0.464 + 0.885i)17-s + (−0.981 + 0.189i)19-s + (−0.377 − 0.925i)20-s + (−0.623 + 0.781i)22-s + (0.938 + 0.346i)25-s + (0.0882 + 0.996i)26-s + (0.999 + 0.0407i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.976 + 0.216i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.976 + 0.216i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2191085764 + 2.001158707i\)
\(L(\frac12)\) \(\approx\) \(0.2191085764 + 2.001158707i\)
\(L(1)\) \(\approx\) \(1.181818263 + 0.8193005209i\)
\(L(1)\) \(\approx\) \(1.181818263 + 0.8193005209i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.876 + 0.482i)T \)
5 \( 1 + (-0.984 - 0.175i)T \)
11 \( 1 + (-0.169 + 0.985i)T \)
13 \( 1 + (0.557 + 0.830i)T \)
17 \( 1 + (0.464 + 0.885i)T \)
19 \( 1 + (-0.981 + 0.189i)T \)
29 \( 1 + (0.999 + 0.0407i)T \)
31 \( 1 + (0.723 + 0.690i)T \)
37 \( 1 + (0.912 + 0.409i)T \)
41 \( 1 + (-0.339 - 0.940i)T \)
43 \( 1 + (0.0611 - 0.998i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (0.288 - 0.957i)T \)
59 \( 1 + (-0.777 - 0.628i)T \)
61 \( 1 + (-0.906 + 0.421i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
71 \( 1 + (0.452 + 0.891i)T \)
73 \( 1 + (0.704 - 0.709i)T \)
79 \( 1 + (-0.928 - 0.371i)T \)
83 \( 1 + (-0.523 - 0.852i)T \)
89 \( 1 + (-0.951 + 0.307i)T \)
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

−18.54323567373341703894653243657, −18.24774277532288881489251830657, −16.82292344025172203065800409848, −16.2653299410095249864702132993, −15.46160463029834925376559950659, −15.16909671184708206355400970253, −14.21347375248331381759526256990, −13.63452505754810963579946197095, −12.88232816137165968957080575357, −12.24476924654953381719216603819, −11.44315921459019540595164213216, −11.03929040250737783123656263121, −10.36431879477657073949143995696, −9.489502812272493779673246683951, −8.371475378847011783815364585636, −7.905376006967864837806890859982, −6.869350916974407792543686959028, −6.18763744579312452609394677229, −5.44238242839328796010639157548, −4.53222904927393709339001655994, −3.958564283618480605674956109746, −2.88591815897953995073370372999, −2.8279693670574363549427259910, −1.215691127735226829848286923075, −0.45774386809134198565044387907, 1.387078657950853340823812390828, 2.32507074529832137924406045865, 3.31730095773429309952903709332, 4.121021667332827136709746901480, 4.48450616371851919506192396931, 5.3427938943667029557317022595, 6.42650963350922414848380030668, 6.80540180396908781661555283665, 7.751178059982505300930767005397, 8.29474822275573950044296526218, 8.95076529523398326059642228348, 10.196373324232353100062367824896, 10.91367700033370895480297338749, 11.7001408430975378168288747869, 12.39033493095658791731945252861, 12.68548760643907731444459842608, 13.66851482878200702896921508934, 14.40533372005246993132987111695, 15.03371960415417411345263830441, 15.61205330036563241482495829600, 16.16050361465833354350119335166, 16.97882139481967261830413123397, 17.44354738105414106236204689750, 18.51462536219102359468307878181, 19.24658526414942020013576047219

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