↑

Properties

Label 1-1911-1911.1706-r0-0-0
Degree $1$
Conductor $1911$
Sign $0.988 + 0.150i$
Analytic cond. $8.87464$
Root an. cond. $8.87464$
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  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (0.222 + 0.974i)8-s + (−0.955 + 0.294i)10-s + (−0.0747 − 0.997i)11-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s + (0.5 − 0.866i)19-s + (−0.988 − 0.149i)20-s + (0.365 − 0.930i)22-s + (−0.623 − 0.781i)23-s + (0.0747 − 0.997i)25-s + (−0.365 − 0.930i)29-s + (0.5 − 0.866i)31-s + (−0.623 + 0.781i)32-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (0.222 + 0.974i)8-s + (−0.955 + 0.294i)10-s + (−0.0747 − 0.997i)11-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s + (0.5 − 0.866i)19-s + (−0.988 − 0.149i)20-s + (0.365 − 0.930i)22-s + (−0.623 − 0.781i)23-s + (0.0747 − 0.997i)25-s + (−0.365 − 0.930i)29-s + (0.5 − 0.866i)31-s + (−0.623 + 0.781i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.988 + 0.150i$
Analytic conductor: \(8.87464\)
Root analytic conductor: \(8.87464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1706, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1911,\ (0:\ ),\ 0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.384604176 + 0.1799831508i\)
\(L(\frac12)\) \(\approx\) \(2.384604176 + 0.1799831508i\)
\(L(1)\) \(\approx\) \(1.561640058 + 0.3730187924i\)
\(L(1)\) \(\approx\) \(1.561640058 + 0.3730187924i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.733 + 0.680i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (0.955 - 0.294i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (-0.365 - 0.930i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.365 + 0.930i)T \)
73 \( 1 + (-0.0747 + 0.997i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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

−19.95038367536458053901743600276, −19.72837857695819624225903775571, −18.79935284217454202813033230305, −17.96534566087566375692576845226, −16.898795360711941639221033261634, −16.19655604046204472769964163811, −15.52914230322597952419478784634, −14.84212934347386185035762485242, −14.17800234990726765698403260525, −13.21853974035417934970850316820, −12.483890461661511863820864577425, −12.12194002905652136467853831183, −11.37572981775834713603757719975, −10.36977815508958090923225080763, −9.80261864693445641934193318008, −8.79582594803861814342411762567, −7.71816718017315834495639295562, −7.21910464184256011110702690766, −6.01177689222580656859087785879, −5.33814872731499593843829802502, −4.48858698157974414539560820803, −3.8499183450747848483982669032, −3.07016708608475843092680400008, −1.80630353323050316059037251598, −1.152043694650381619105733546442, 0.66540412026522348442342623772, 2.515112762093011717173440643582, 2.897089623431371909823652812430, 3.93209154724039351333934341882, 4.49803470626412045495355490756, 5.68839295561350732952322098093, 6.18324355771353192059857977909, 7.22757797084275348547210924922, 7.70453937408924840668895022756, 8.47875178204800107700298806872, 9.56185202604901528571252583595, 10.73722585823118280766925886994, 11.33719420476943441997209522017, 11.86923263014539334128987157760, 12.732661342639337055138746477032, 13.66867074357977411408211005398, 14.164498813828565452563740067890, 14.873226456016648990383955895278, 15.72759776181683597142792096749, 16.11660098057436501536696185590, 16.89199433395660857239185954563, 17.88299728472030685698412094437, 18.63804885061736011967406542780, 19.40362012016268202521372817498, 20.162507971024333444369625293998

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