↑

Properties

Label 1-39e2-1521.743-r0-0-0
Degree $1$
Conductor $1521$
Sign $-0.902 + 0.430i$
Analytic cond. $7.06349$
Root an. cond. $7.06349$
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  + (0.903 + 0.428i)2-s + (0.632 + 0.774i)4-s + (−0.316 + 0.948i)5-s + (−0.239 − 0.970i)7-s + (0.239 + 0.970i)8-s + (−0.692 + 0.721i)10-s + (−0.0804 + 0.996i)11-s + (0.200 − 0.979i)14-s + (−0.200 + 0.979i)16-s + (0.692 + 0.721i)17-s + (−0.866 + 0.5i)19-s + (−0.935 + 0.354i)20-s + (−0.5 + 0.866i)22-s + 23-s + (−0.799 − 0.600i)25-s + ⋯
L(s)  = 1  + (0.903 + 0.428i)2-s + (0.632 + 0.774i)4-s + (−0.316 + 0.948i)5-s + (−0.239 − 0.970i)7-s + (0.239 + 0.970i)8-s + (−0.692 + 0.721i)10-s + (−0.0804 + 0.996i)11-s + (0.200 − 0.979i)14-s + (−0.200 + 0.979i)16-s + (0.692 + 0.721i)17-s + (−0.866 + 0.5i)19-s + (−0.935 + 0.354i)20-s + (−0.5 + 0.866i)22-s + 23-s + (−0.799 − 0.600i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-0.902 + 0.430i$
Analytic conductor: \(7.06349\)
Root analytic conductor: \(7.06349\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1521,\ (0:\ ),\ -0.902 + 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4479593550 + 1.979458065i\)
\(L(\frac12)\) \(\approx\) \(0.4479593550 + 1.979458065i\)
\(L(1)\) \(\approx\) \(1.262244014 + 0.8710982769i\)
\(L(1)\) \(\approx\) \(1.262244014 + 0.8710982769i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.903 + 0.428i)T \)
5 \( 1 + (-0.316 + 0.948i)T \)
7 \( 1 + (-0.239 - 0.970i)T \)
11 \( 1 + (-0.0804 + 0.996i)T \)
17 \( 1 + (0.692 + 0.721i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.428 + 0.903i)T \)
31 \( 1 + (-0.600 - 0.799i)T \)
37 \( 1 + (0.391 - 0.919i)T \)
41 \( 1 + (-0.464 + 0.885i)T \)
43 \( 1 + (-0.120 + 0.992i)T \)
47 \( 1 + (0.160 - 0.987i)T \)
53 \( 1 + (0.970 - 0.239i)T \)
59 \( 1 + (-0.979 + 0.200i)T \)
61 \( 1 + (-0.970 - 0.239i)T \)
67 \( 1 + (-0.935 + 0.354i)T \)
71 \( 1 + (0.999 - 0.0402i)T \)
73 \( 1 + (0.822 + 0.568i)T \)
79 \( 1 + (0.987 + 0.160i)T \)
83 \( 1 + (-0.999 - 0.0402i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.663 + 0.748i)T \)
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

−20.48790822381044039581451984467, −19.51525197569071068996883546427, −19.04474742560918632623475974568, −18.37028793307475357550937749837, −16.93643582970436553571436715904, −16.450868827205367522099294553198, −15.467921861499443045057349922603, −15.21606391098210853239520305668, −14.01515729261971837948167095226, −13.372844021313593728540914634174, −12.62148164466924773304289885791, −12.03884342011465721399017022424, −11.364897913579840606561870626321, −10.55602647078436111839975521492, −9.36585863793222497265638438164, −8.89262697244501762922005567806, −7.88084503070136637475740671478, −6.770352279776614365469747606583, −5.80182125221421382684197419919, −5.27810138049519577970287932427, −4.48839083802374790516840879258, −3.43338315508064176086678283912, −2.741383098891394093285183916651, −1.65064449122722999756175961007, −0.52055504008809266517266276349, 1.63178090009183505786098593391, 2.674253591925893702154073346307, 3.650976613706947784475388003060, 4.09435672790478339349865577232, 5.12054569679959617909778147810, 6.19510862116570742570592088878, 6.853252922481673097531279241302, 7.4989105770333696481367079382, 8.097278713083763055602187932955, 9.49275174899189229369426687912, 10.56010183672023760504034815117, 10.88712481806343229245382815182, 11.953688443290693067932605552826, 12.76728964507580185423258149587, 13.3125810371853635751501074968, 14.381438444200038399431542264151, 14.819241433564442422834594948483, 15.311322442453400568778919437337, 16.53400591684983183436624417927, 16.85607935438548416206770886313, 17.808109928686713198973303617289, 18.653995207907787997716561909797, 19.686236460306645794153239764263, 20.14127601481301247759659325569, 21.140273987522045015807159161811

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