↑

Properties

Label 1-43e2-1849.1012-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.138 + 0.990i$
Analytic cond. $8.58671$
Root an. cond. $8.58671$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0677 + 0.997i)2-s + (−0.999 + 0.0173i)3-s + (−0.990 + 0.135i)4-s + (0.976 − 0.214i)5-s + (−0.0851 − 0.996i)6-s + (0.435 − 0.900i)7-s + (−0.202 − 0.979i)8-s + (0.999 − 0.0347i)9-s + (0.279 + 0.960i)10-s + (0.973 − 0.227i)11-s + (0.988 − 0.152i)12-s + (0.725 + 0.687i)13-s + (0.927 + 0.373i)14-s + (−0.972 + 0.230i)15-s + (0.963 − 0.268i)16-s + (0.584 + 0.811i)17-s + ⋯
L(s)  = 1  + (0.0677 + 0.997i)2-s + (−0.999 + 0.0173i)3-s + (−0.990 + 0.135i)4-s + (0.976 − 0.214i)5-s + (−0.0851 − 0.996i)6-s + (0.435 − 0.900i)7-s + (−0.202 − 0.979i)8-s + (0.999 − 0.0347i)9-s + (0.279 + 0.960i)10-s + (0.973 − 0.227i)11-s + (0.988 − 0.152i)12-s + (0.725 + 0.687i)13-s + (0.927 + 0.373i)14-s + (−0.972 + 0.230i)15-s + (0.963 − 0.268i)16-s + (0.584 + 0.811i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.138 + 0.990i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (1012, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246485491 + 1.083839586i\)
\(L(\frac12)\) \(\approx\) \(1.246485491 + 1.083839586i\)
\(L(1)\) \(\approx\) \(0.9729987172 + 0.4879251908i\)
\(L(1)\) \(\approx\) \(0.9729987172 + 0.4879251908i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.0677 + 0.997i)T \)
3 \( 1 + (-0.999 + 0.0173i)T \)
5 \( 1 + (0.976 - 0.214i)T \)
7 \( 1 + (0.435 - 0.900i)T \)
11 \( 1 + (0.973 - 0.227i)T \)
13 \( 1 + (0.725 + 0.687i)T \)
17 \( 1 + (0.584 + 0.811i)T \)
19 \( 1 + (0.0747 + 0.997i)T \)
23 \( 1 + (0.899 + 0.437i)T \)
29 \( 1 + (-0.756 + 0.654i)T \)
31 \( 1 + (0.990 + 0.138i)T \)
37 \( 1 + (0.478 - 0.877i)T \)
41 \( 1 + (-0.993 - 0.114i)T \)
47 \( 1 + (-0.529 + 0.848i)T \)
53 \( 1 + (0.949 - 0.314i)T \)
59 \( 1 + (-0.882 + 0.471i)T \)
61 \( 1 + (-0.974 - 0.224i)T \)
67 \( 1 + (0.358 + 0.933i)T \)
71 \( 1 + (0.650 + 0.759i)T \)
73 \( 1 + (-0.208 + 0.977i)T \)
79 \( 1 + (0.711 + 0.702i)T \)
83 \( 1 + (0.681 - 0.731i)T \)
89 \( 1 + (0.706 + 0.707i)T \)
97 \( 1 + (-0.999 + 0.0104i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.139583800773981594117496929345, −18.95669883879586297063186175596, −18.4461024794927410303477397103, −17.94675218941022711731523853731, −17.233147799293031851486317538965, −16.70644009438972944662772462489, −15.30926313722578305685143759597, −14.856253655226712788985329969560, −13.62793145380923377819622362709, −13.30371033136589383332353808005, −12.25816035114392202640729458513, −11.765282398325008512308160430629, −11.072777907936403032206784058908, −10.39531995886326951556552771372, −9.49092993459599002107645828047, −9.09702365670490609136736508871, −7.96660202471725124766883263744, −6.6467651323576048411082291366, −6.01392142255044420141945579419, −5.12380216874729976949300170576, −4.75091779768070183762921123084, −3.410494997823668139335738353539, −2.52404025568872369557236218320, −1.579833850160427984071695063134, −0.835935707090045298419012749932, 1.19604234523841020738539404331, 1.42085032200457409118767029170, 3.62379867111554835636861911476, 4.19328056649874538551069062296, 5.12963037806244370318302446417, 5.82288302252204717117954227949, 6.46954822252664828082166926252, 7.05625107966794828299522260952, 8.05609347044221746142770678403, 8.98960387983701985052016811152, 9.74882357347007124026599417762, 10.46285005880744267509010711856, 11.27880974564075399993156334449, 12.289488787477128951090147981630, 13.0399471029557577757795303028, 13.77595538869486359566891058099, 14.34249361232842544946302990757, 15.117575702279260693861472370421, 16.4060018977656289562495022449, 16.6121456602325796622695967667, 17.20730075517080366819282424014, 17.74706508255328761187423098110, 18.56498866896362536090368569658, 19.219281621831467035368764924, 20.55945862358509985070031229507

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