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Properties

Label 1-3381-3381.1511-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.865 + 0.500i$
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  + (−0.996 − 0.0815i)2-s + (0.986 + 0.162i)4-s + (0.182 + 0.983i)5-s + (−0.970 − 0.242i)8-s + (−0.101 − 0.994i)10-s + (0.933 + 0.359i)11-s + (0.714 − 0.699i)13-s + (0.947 + 0.320i)16-s + (0.986 − 0.162i)17-s + (0.959 − 0.281i)19-s + (0.0203 + 0.999i)20-s + (−0.900 − 0.433i)22-s + (−0.933 + 0.359i)25-s + (−0.768 + 0.639i)26-s + (−0.986 + 0.162i)29-s + ⋯
L(s)  = 1  + (−0.996 − 0.0815i)2-s + (0.986 + 0.162i)4-s + (0.182 + 0.983i)5-s + (−0.970 − 0.242i)8-s + (−0.101 − 0.994i)10-s + (0.933 + 0.359i)11-s + (0.714 − 0.699i)13-s + (0.947 + 0.320i)16-s + (0.986 − 0.162i)17-s + (0.959 − 0.281i)19-s + (0.0203 + 0.999i)20-s + (−0.900 − 0.433i)22-s + (−0.933 + 0.359i)25-s + (−0.768 + 0.639i)26-s + (−0.986 + 0.162i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.865 + 0.500i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.865 + 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.370662958 + 0.3678940061i\)
\(L(\frac12)\) \(\approx\) \(1.370662958 + 0.3678940061i\)
\(L(1)\) \(\approx\) \(0.8760440602 + 0.1193157325i\)
\(L(1)\) \(\approx\) \(0.8760440602 + 0.1193157325i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.996 - 0.0815i)T \)
5 \( 1 + (0.182 + 0.983i)T \)
11 \( 1 + (0.933 + 0.359i)T \)
13 \( 1 + (0.714 - 0.699i)T \)
17 \( 1 + (0.986 - 0.162i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (-0.986 + 0.162i)T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + (0.917 + 0.396i)T \)
41 \( 1 + (0.182 + 0.983i)T \)
43 \( 1 + (0.970 - 0.242i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (0.992 - 0.122i)T \)
59 \( 1 + (0.101 + 0.994i)T \)
61 \( 1 + (0.768 + 0.639i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.301 - 0.953i)T \)
73 \( 1 + (-0.488 - 0.872i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (-0.591 + 0.806i)T \)
89 \( 1 + (-0.979 - 0.202i)T \)
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

−18.833276311706880697697401500536, −18.02221689937154278234747153473, −17.323392873987048146541448140349, −16.63065602075752079428681127637, −16.29541616628968637922056065842, −15.68210827440255748448315679565, −14.53760809834567737254705706432, −14.11638771845281212572761996199, −13.09285052110116981797278215206, −12.27268487327160737929160156731, −11.68449522984921540577743986068, −11.0998224534925177132475102619, −10.08073065119834756684902022054, −9.43284449014708405983195220625, −8.93104918739830468410130170703, −8.30180135825700306576419725716, −7.497505757755861682907051776526, −6.73327470436598637033368891212, −5.774439199879094727702982343248, −5.42734369583562084258180331579, −4.03748567826976594829710292052, −3.46772271016430981499944486714, −2.177084929666720109503246832688, −1.35905685985126302641158031490, −0.83564591309768524634275656847, 0.86918085091395154938987892844, 1.63696684907691881585511182987, 2.67887131864958374503658539256, 3.28268949647584806230106620156, 4.067230325832939796855131132549, 5.61301435550969492586653598555, 6.036818107418821364314237740616, 6.9486377923304593326513354648, 7.53706124265592369437501830137, 8.11355039962620862974867933783, 9.29597395559238041028123900334, 9.590664470731890850803938496479, 10.381793062164625906895264300529, 11.15175068147450022693129743029, 11.56685324274056482481596310045, 12.399258043744068803330479325319, 13.32005566159108452985845988090, 14.20361261494761182619986096928, 14.95709607899089586688361072650, 15.32515279954299366086387403651, 16.40186867377861505560853304364, 16.77966500866702398939344696417, 17.85778091915443223053941681225, 18.04765079497611953143973626841, 18.77267954157507282569569616346

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