Properties

Degree 1
Conductor $ 13 \cdot 31 $
Sign $0.0168 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.994 − 0.104i)2-s + (0.104 − 0.994i)3-s + (0.978 − 0.207i)4-s + (0.866 − 0.5i)5-s i·6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.978 − 0.207i)9-s + (0.809 − 0.587i)10-s + (0.951 + 0.309i)11-s + (−0.104 − 0.994i)12-s + (−0.978 − 0.207i)14-s + (−0.406 − 0.913i)15-s + (0.913 − 0.406i)16-s + (0.309 + 0.951i)17-s + (−0.994 − 0.104i)18-s + ⋯
L(s,χ)  = 1  + (0.994 − 0.104i)2-s + (0.104 − 0.994i)3-s + (0.978 − 0.207i)4-s + (0.866 − 0.5i)5-s i·6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.978 − 0.207i)9-s + (0.809 − 0.587i)10-s + (0.951 + 0.309i)11-s + (−0.104 − 0.994i)12-s + (−0.978 − 0.207i)14-s + (−0.406 − 0.913i)15-s + (0.913 − 0.406i)16-s + (0.309 + 0.951i)17-s + (−0.994 − 0.104i)18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0168 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0168 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.0168 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{403} (24, \cdot )$
Sato-Tate  :  $\mu(60)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 403,\ (0:\ ),\ 0.0168 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.900018234 - 1.932356242i$
$L(\frac12,\chi)$  $\approx$  $1.900018234 - 1.932356242i$
$L(\chi,1)$  $\approx$  1.802374646 - 0.9932658418i
$L(1,\chi)$  $\approx$  1.802374646 - 0.9932658418i

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

−24.959112496727890913890610990741, −23.31112691476472667977933865893, −22.59818274390737813545693257466, −21.9907401722166619305272712408, −21.460625186447642199470148761094, −20.49984433475418359375275278102, −19.673642317674598512312009965149, −18.60216498085848335394835451651, −17.09291367468182602184891574449, −16.54341590257294406869867367608, −15.65086055814226770404749506301, −14.74503983896412489224444704945, −14.06580945748266543298862755765, −13.313814807443051902176303670, −12.044455471321727622572788816167, −11.2586178095376685056233964881, −10.02229858015598006458371727722, −9.62271301316481587376373741147, −8.20864902132962353613643684065, −6.5939693833188473113916498494, −6.085108265552407585256000659553, −5.11892122054008689159282287279, −3.8425618112530182090273371438, −3.13999331918138751543876788026, −2.084539406190434790762380182009, 1.24421728039599331177419826786, 2.17362599360826347807129755048, 3.32960114117137747644856686226, 4.52374337027819740422552374857, 5.948426278068327630511178080545, 6.35533446229563440168536808263, 7.27103899789314154459459399391, 8.65253217326789049070521668182, 9.75048238018908095599767711809, 10.831523896423945899290017805434, 12.14421423299977785209254095146, 12.67033219201404396332541304622, 13.373737180075465880164664882185, 14.1281339788618399717773352281, 14.95301134443155422067366586090, 16.38175854384208809698902581576, 16.98798125420450784709026546978, 17.952507115512223769047003861295, 19.3438101773580952279840543254, 19.78181967318441593109428681331, 20.59503922913353722745047793070, 21.825549235586392832405847204507, 22.325582364824076829419955353677, 23.44868317030470442267623524918, 23.99286886542146750695674641063

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