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Properties

Label 1-3381-3381.1310-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.981 - 0.191i$
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  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + (0.222 − 0.974i)8-s + (0.222 + 0.974i)10-s + (−0.900 + 0.433i)11-s + (−0.900 + 0.433i)13-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s − 19-s + (0.623 + 0.781i)20-s + (−0.623 + 0.781i)22-s + (−0.900 − 0.433i)25-s + (−0.623 + 0.781i)26-s + (−0.623 − 0.781i)29-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + (0.222 − 0.974i)8-s + (0.222 + 0.974i)10-s + (−0.900 + 0.433i)11-s + (−0.900 + 0.433i)13-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s − 19-s + (0.623 + 0.781i)20-s + (−0.623 + 0.781i)22-s + (−0.900 − 0.433i)25-s + (−0.623 + 0.781i)26-s + (−0.623 − 0.781i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.981 - 0.191i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1310, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.981 - 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04738742826 - 0.4912202657i\)
\(L(\frac12)\) \(\approx\) \(0.04738742826 - 0.4912202657i\)
\(L(1)\) \(\approx\) \(1.240766151 - 0.2126613527i\)
\(L(1)\) \(\approx\) \(1.240766151 - 0.2126613527i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 - T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.623 - 0.781i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
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

−19.275008815110274417167539846296, −18.439339769892000770064283782727, −17.39630722691624592283549298159, −16.94286248020058773884154049155, −16.28956487496585295830310272826, −15.64771025096199162916465492214, −15.08387194866057228180105331103, −14.27106234550933205463755692670, −13.47291279854116593710789839837, −12.98938997942924651193194784409, −12.2474251027141581824383585012, −11.84857827994088967390653422955, −10.83758170631018382132693480232, −10.115281683972807467036644573598, −9.04041749143048320337400645811, −8.33379526699792472197575679255, −7.69658226165595208899249418770, −7.088470197472178503404063504194, −5.97927341813562836525882171085, −5.35772890335078060140837057272, −4.80119355046289117651994906940, −4.10913886834671190224248790540, −3.05610495949202408808002904561, −2.484989407612605795192732817962, −1.28144541632441594485304872490, 0.09383942829141797444140794823, 1.72602937094765066922963513668, 2.40398104970233399045394544516, 2.990988896940992355402823243961, 4.03256726292018958126463183588, 4.46906462272342279456729790685, 5.58893171433010769296201306143, 6.08450056301546805443688167509, 7.12063468349631962441862926889, 7.43546370253369531202945242145, 8.50028626843611976429609623893, 9.76462280066260932660562510816, 10.23775273846694535025197156052, 10.78566372368077655966341667062, 11.58128789259694663878381460539, 12.27811857398900177921907616365, 12.840284724066844776958433831315, 13.68920358542847450720711562971, 14.35899703882838104776524306792, 15.0238440475940718226994142049, 15.334058267577199439653486280962, 16.2216576516771218283595739558, 17.14009448524653051070345345655, 17.88269233148020454730416223134, 18.92617650844671221388892897297

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