Properties

Degree 1
Conductor $ 7 \cdot 11 \cdot 13 $
Sign $0.949 + 0.313i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 − 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s − 8-s + 9-s − 10-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s − 19-s + (−0.5 + 0.866i)20-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s − 8-s + 9-s − 10-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s − 19-s + (−0.5 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.949 + 0.313i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.949 + 0.313i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1001\)    =    \(7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $0.949 + 0.313i$
motivic weight  =  \(0\)
character  :  $\chi_{1001} (373, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1001,\ (1:\ ),\ 0.949 + 0.313i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.813214441 + 0.2914647718i$
$L(\frac12,\chi)$  $\approx$  $1.813214441 + 0.2914647718i$
$L(\chi,1)$  $\approx$  1.266335248 - 0.6687465344i
$L(1,\chi)$  $\approx$  1.266335248 - 0.6687465344i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.36129398806025920405887258838, −20.895282893065472942357122275026, −19.810318213756911698271678311072, −18.95877430176926644677020249736, −18.38595659297616573182189150241, −17.5145335599743365174326117894, −16.33461715834572407987692301698, −15.74869706255436967064661961572, −14.973934656559360175491677202533, −14.3708074857971089617795574572, −13.86613297145340404648810932646, −12.84287137476778258630175014474, −12.13547556459774639277546833913, −11.01577903450867951863258753253, −9.94207891513244873149190099804, −9.06233829295210764017390138780, −8.10174924470149965374539048153, −7.592969605345757435935555883563, −6.77454018176398885917832138814, −5.96258750223932671110333628354, −4.54626855531918272654204486034, −3.954875947394418868583994054335, −2.995060599581380365728007509889, −2.27924155195292677647654346441, −0.27136938616010157246375468317, 1.2205981172203777711213885710, 1.85093618807918152018032807203, 3.11824399604967639265725205484, 3.821365343929635052103579214031, 4.56954007657365407449131659421, 5.4907155631845629620899559748, 6.72681873801776018323228552733, 8.00606079406041429630122757114, 8.60775457392469546225107044652, 9.386348329119969101037081040, 10.233844277270116454912806785154, 11.09130954065440411573550827797, 12.354163319434802470207164847078, 12.55157181322303744055103838590, 13.50005092553581585752736523288, 14.20952662051361204537174846743, 15.08894492123415742393085298215, 15.64728388065589388267285574230, 16.71197393972006643168216553953, 17.80273035545717409063910599795, 18.82117479936142931547704457006, 19.44601085247605057265457247594, 19.97100492190018743034443008895, 20.61591024802205065515301197657, 21.48289975470052840560759863323

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