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Properties

Label 1-3381-3381.1229-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.530 + 0.847i$
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  + (0.973 − 0.229i)2-s + (0.894 − 0.446i)4-s + (−0.966 − 0.255i)5-s + (0.768 − 0.639i)8-s + (−0.999 − 0.0271i)10-s + (−0.00679 + 0.999i)11-s + (0.101 − 0.994i)13-s + (0.601 − 0.798i)16-s + (−0.833 + 0.552i)17-s + (−0.723 + 0.690i)19-s + (−0.979 + 0.202i)20-s + (0.222 + 0.974i)22-s + (0.869 + 0.494i)25-s + (−0.128 − 0.991i)26-s + (0.0611 + 0.998i)29-s + ⋯
L(s)  = 1  + (0.973 − 0.229i)2-s + (0.894 − 0.446i)4-s + (−0.966 − 0.255i)5-s + (0.768 − 0.639i)8-s + (−0.999 − 0.0271i)10-s + (−0.00679 + 0.999i)11-s + (0.101 − 0.994i)13-s + (0.601 − 0.798i)16-s + (−0.833 + 0.552i)17-s + (−0.723 + 0.690i)19-s + (−0.979 + 0.202i)20-s + (0.222 + 0.974i)22-s + (0.869 + 0.494i)25-s + (−0.128 − 0.991i)26-s + (0.0611 + 0.998i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.530 + 0.847i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.530 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3033023048 + 0.5476748407i\)
\(L(\frac12)\) \(\approx\) \(0.3033023048 + 0.5476748407i\)
\(L(1)\) \(\approx\) \(1.277431614 - 0.1511710947i\)
\(L(1)\) \(\approx\) \(1.277431614 - 0.1511710947i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.973 - 0.229i)T \)
5 \( 1 + (-0.966 - 0.255i)T \)
11 \( 1 + (-0.00679 + 0.999i)T \)
13 \( 1 + (0.101 - 0.994i)T \)
17 \( 1 + (-0.833 + 0.552i)T \)
19 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (0.0611 + 0.998i)T \)
31 \( 1 + (-0.995 + 0.0950i)T \)
37 \( 1 + (-0.994 + 0.108i)T \)
41 \( 1 + (-0.262 - 0.965i)T \)
43 \( 1 + (0.768 + 0.639i)T \)
47 \( 1 + (-0.365 - 0.930i)T \)
53 \( 1 + (-0.984 + 0.175i)T \)
59 \( 1 + (-0.999 - 0.0271i)T \)
61 \( 1 + (0.923 - 0.384i)T \)
67 \( 1 + (-0.928 + 0.371i)T \)
71 \( 1 + (-0.996 + 0.0815i)T \)
73 \( 1 + (-0.612 + 0.790i)T \)
79 \( 1 + (-0.0475 - 0.998i)T \)
83 \( 1 + (-0.999 - 0.0407i)T \)
89 \( 1 + (-0.546 - 0.837i)T \)
97 \( 1 + (0.654 + 0.755i)T \)
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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

−18.86952746948833154394531390166, −17.81555656406265648518902389700, −16.92120619613224417020342940414, −16.282703638091295525047177952, −15.7481146996606545886448472156, −15.16988049567973044635975190925, −14.36930823234842081980091490715, −13.79795335963293084817360393512, −13.11419989610385724619126346688, −12.335046598439628401448714460953, −11.456553716814316474678082994563, −11.26824042650056540024546522124, −10.55043825920611301864891760549, −9.19107369465387164870370596474, −8.537309235204331290286935539054, −7.75027315890276186750400836753, −6.96541474316098365857260809418, −6.46698189925476066583720649388, −5.62882019254117601854957767291, −4.52312939542336344228583588398, −4.24668284575395856314873095070, −3.28713831228093654427181557739, −2.65410608215024318970918873743, −1.64690224231053127346413953134, −0.11734424481509663376595712765, 1.35957675075378231199062149931, 2.110608566780398809159733149669, 3.18704826840663828528877939248, 3.80671576915677559394782501825, 4.519346147952323552383178721393, 5.16470016833704428830473195933, 6.0165913673193650048326132854, 6.94302415055011164205432403275, 7.47241030912170883407994401913, 8.31410293752431537491880079113, 9.12745480108770888772675420669, 10.42154464066571016058856417134, 10.58264965300949518477042061924, 11.537734168962378987393295307648, 12.27315485395982156417892101443, 12.77831883335667682662998901497, 13.17991713410061613417722914110, 14.50201794859160529974782899574, 14.73437043566193045768149353956, 15.6482154270549888784115529881, 15.85504123789198983367340582697, 16.88643955518764829608879648460, 17.58030472814958318909596676899, 18.561488855342249077893500333278, 19.2813785495720550986081204101

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