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Properties

Label 1-3381-3381.911-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.803 - 0.595i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.803 - 0.595i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.803 - 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1855818134 - 0.5616102246i\)
\(L(\frac12)\) \(\approx\) \(0.1855818134 - 0.5616102246i\)
\(L(1)\) \(\approx\) \(0.6646143563 - 0.1801789570i\)
\(L(1)\) \(\approx\) \(0.6646143563 - 0.1801789570i\)

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

−18.945586369632405411753735193911, −18.50695347212247695561113374158, −17.57940243729246316508235114889, −17.08175556956801577676091240381, −16.669074260660404230396457330225, −15.637437869248616008485576489159, −15.00647648159866616903137218121, −14.398225687667710803735222259296, −13.76481807410037960442796999860, −12.87201492819409919096873623405, −11.8300095386629561443018507343, −11.24747771915975891929271554941, −10.36873300144119525439166109206, −10.08544242753257682794538723561, −9.1582003941706615197365851074, −8.59860507758284850361102758178, −7.68862622038866266461591529879, −7.00219931926246187838672676253, −6.34407214364550925241367632113, −5.65379804330044658524714440540, −4.96367882526372028500997083502, −3.50891535101772016119701791259, −2.7593825942188121567569460056, −2.00304639622537091039900591163, −1.09451963833752152685124090999, 0.23788815143416802853400039068, 1.50818733468559634682523173735, 1.971030628075356818375056808504, 2.8386553963801202123344437230, 3.88659898108170676653877383580, 4.83776178059633441638753072126, 5.61571101008670513612634657897, 6.50284638169136076257298114795, 7.36669788690424290519467515333, 7.88578808826927460293033080182, 8.80149538614680123046831299447, 9.57723718611578726956884338849, 9.86538993099870072180999743499, 10.53510691287686082689425594442, 11.644014026471431537584813326568, 12.20737470500603843172345235595, 12.75607366892461720845404589694, 13.466252518140165674548128833833, 14.727644376374910376536559681871, 14.9082176726904316183620643710, 16.25096782426430611085526757236, 16.612411811163415729413875193832, 17.15458149336531049599476853998, 17.94586597968695479286109799976, 18.35142415688725250115662276944

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