Properties

Label 1-3381-3381.1241-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.949 - 0.315i$
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  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.365 − 0.930i)5-s + (−0.623 − 0.781i)8-s + (−0.365 − 0.930i)10-s + (0.955 + 0.294i)11-s + (−0.222 − 0.974i)13-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (0.5 + 0.866i)19-s + (−0.900 − 0.433i)20-s + (0.900 − 0.433i)22-s + (−0.733 − 0.680i)25-s + (−0.826 − 0.563i)26-s + (0.900 + 0.433i)29-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.365 − 0.930i)5-s + (−0.623 − 0.781i)8-s + (−0.365 − 0.930i)10-s + (0.955 + 0.294i)11-s + (−0.222 − 0.974i)13-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (0.5 + 0.866i)19-s + (−0.900 − 0.433i)20-s + (0.900 − 0.433i)22-s + (−0.733 − 0.680i)25-s + (−0.826 − 0.563i)26-s + (0.900 + 0.433i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.949 - 0.315i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.949 - 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4574874846 - 2.829721854i\)
\(L(\frac12)\) \(\approx\) \(0.4574874846 - 2.829721854i\)
\(L(1)\) \(\approx\) \(1.250683799 - 1.181179827i\)
\(L(1)\) \(\approx\) \(1.250683799 - 1.181179827i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.733 - 0.680i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
11 \( 1 + (0.955 + 0.294i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.826 - 0.563i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.0747 - 0.997i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (0.0747 - 0.997i)T \)
59 \( 1 + (-0.365 - 0.930i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.733 - 0.680i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
97 \( 1 - T \)
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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

−18.975918888202913343058939355866, −18.44469804778808575504393650594, −17.45105221109471276706716644697, −17.04850090896992366918091735948, −16.39722475807581181183179495093, −15.47529131190868454728804082771, −14.87983121966686929339781750290, −14.301808348054049448469358137225, −13.74379850548415714101242325423, −13.196722980919899151104683942109, −11.92346993187056751914763765774, −11.79553274004411690313216264412, −10.88456097255841900428091562909, −9.90865410695030221317994052517, −9.18705161263183662516890463658, −8.39891619314504640344955175812, −7.433324176451963440337586512343, −6.89755364302779293015017730779, −6.23749303251088050887446385605, −5.674355164487526460140508122571, −4.62656984190533592518228862832, −3.927501558190077737440829376689, −3.14303071091256658982263625568, −2.43595779733777922682986506352, −1.364339113955608651138545016180, 0.6572746662460882144954934798, 1.40670414785127951800569615682, 2.15341739496985414505940654434, 3.316141840541310895793611550573, 3.77379580931081986861102320718, 4.97401241608623411389980682406, 5.19317538452165819475965068732, 6.05482663142346285599587340410, 6.885346115164578454699377878497, 7.89340605405828508406766367299, 8.790640528297359690828411111782, 9.57643208971032659257619913123, 10.02137116748885807567570005677, 10.82527435089792884187671064450, 11.82260083447561858925123263037, 12.3810549232423632340365558263, 12.64589880480022485861591200676, 13.71807403825316560228799427142, 14.161646858507378437871952538332, 14.812586874670345508219236244390, 15.747403701918196996807701550984, 16.33453484959551838782746215369, 17.13077050244726525846347397155, 17.92356016748851389344258982926, 18.54086754536874481001659945339

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