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Properties

Label 1-2011-2011.19-r1-0-0
Degree $1$
Conductor $2011$
Sign $0.993 - 0.111i$
Analytic cond. $216.111$
Root an. cond. $216.111$
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  + (0.0546 − 0.998i)2-s + (−0.880 − 0.474i)3-s + (−0.994 − 0.109i)4-s + (−0.999 + 0.0406i)5-s + (−0.521 + 0.853i)6-s + (−0.964 − 0.265i)7-s + (−0.163 + 0.986i)8-s + (0.550 + 0.834i)9-s + (−0.0140 + 0.999i)10-s + (0.995 + 0.0967i)11-s + (0.823 + 0.567i)12-s + (−0.537 + 0.843i)13-s + (−0.317 + 0.948i)14-s + (0.899 + 0.437i)15-s + (0.976 + 0.217i)16-s + (0.166 − 0.986i)17-s + ⋯
L(s)  = 1  + (0.0546 − 0.998i)2-s + (−0.880 − 0.474i)3-s + (−0.994 − 0.109i)4-s + (−0.999 + 0.0406i)5-s + (−0.521 + 0.853i)6-s + (−0.964 − 0.265i)7-s + (−0.163 + 0.986i)8-s + (0.550 + 0.834i)9-s + (−0.0140 + 0.999i)10-s + (0.995 + 0.0967i)11-s + (0.823 + 0.567i)12-s + (−0.537 + 0.843i)13-s + (−0.317 + 0.948i)14-s + (0.899 + 0.437i)15-s + (0.976 + 0.217i)16-s + (0.166 − 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ 0.993 - 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6985268656 - 0.03891382314i\)
\(L(\frac12)\) \(\approx\) \(0.6985268656 - 0.03891382314i\)
\(L(1)\) \(\approx\) \(0.5145428394 - 0.3089514214i\)
\(L(1)\) \(\approx\) \(0.5145428394 - 0.3089514214i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (0.0546 - 0.998i)T \)
3 \( 1 + (-0.880 - 0.474i)T \)
5 \( 1 + (-0.999 + 0.0406i)T \)
7 \( 1 + (-0.964 - 0.265i)T \)
11 \( 1 + (0.995 + 0.0967i)T \)
13 \( 1 + (-0.537 + 0.843i)T \)
17 \( 1 + (0.166 - 0.986i)T \)
19 \( 1 + (0.876 - 0.482i)T \)
23 \( 1 + (0.953 + 0.301i)T \)
29 \( 1 + (0.707 + 0.706i)T \)
31 \( 1 + (-0.405 - 0.914i)T \)
37 \( 1 + (-0.566 + 0.824i)T \)
41 \( 1 + (0.464 + 0.885i)T \)
43 \( 1 + (-0.227 + 0.973i)T \)
47 \( 1 + (0.976 - 0.214i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.965 + 0.259i)T \)
61 \( 1 + (0.510 + 0.859i)T \)
67 \( 1 + (-0.486 - 0.873i)T \)
71 \( 1 + (-0.997 + 0.0718i)T \)
73 \( 1 + (-0.977 - 0.210i)T \)
79 \( 1 + (0.382 + 0.924i)T \)
83 \( 1 + (-0.917 - 0.398i)T \)
89 \( 1 + (-0.703 - 0.710i)T \)
97 \( 1 + (0.990 - 0.140i)T \)
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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

−19.47080903496490767595681210400, −19.04689003311992809651554367904, −18.14353850888903628055202848386, −17.1934420959419110469058923097, −16.892041829296125851750642058304, −15.97065219128848951310660186324, −15.64006573630548607072044228440, −14.94093968693783261730120602547, −14.22148632829134051025039922671, −12.9780133507677095476630571362, −12.30222421850387720004329622657, −12.00684388391117089339158294239, −10.68579028876468389007712392121, −10.10988419982076955945391581846, −9.13652080048476383662156386544, −8.61401328020896138110134994374, −7.38556870732165292364816438815, −6.96484649187643131461394504004, −6.001441304847943797907794612, −5.482637742110059975813275100476, −4.49503498348344601939793212826, −3.74554921474989199686352905513, −3.193187149735728051582990297496, −1.01414971550544674011257199891, −0.27167393541025839959478116141, 0.688053401504313460693169363747, 1.304455305139503917625944824338, 2.6863715120834668143751364270, 3.40831517334190826443372176064, 4.424000440816173158534022228841, 4.886484296471313662382604375353, 6.06226178504158498658575420687, 7.085236481413316261194284332768, 7.37725777059053336905663236669, 8.77369819316546067566760533495, 9.46336479572969964806708604286, 10.1828401344261063782268779915, 11.21900875104060714552034661698, 11.66455929022126211246233310574, 12.11038338507391943298071182491, 12.90348291776273019533035553609, 13.61524405433139663080735624440, 14.39595804562999444158692055813, 15.42340818128992445666922640458, 16.477792039680040955429340637396, 16.73610878374807144135889272377, 17.732506702153480116106982076550, 18.56437001384676510065467206682, 19.075164529563843149711999600071, 19.729749961377944236634934438426

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