Properties

Degree 1
Conductor 193
Sign $0.981 + 0.189i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.980 + 0.195i)2-s + (−0.382 + 0.923i)3-s + (0.923 + 0.382i)4-s + (0.0980 − 0.995i)5-s + (−0.555 + 0.831i)6-s + (−0.707 − 0.707i)7-s + (0.831 + 0.555i)8-s + (−0.707 − 0.707i)9-s + (0.290 − 0.956i)10-s + (0.773 + 0.634i)11-s + (−0.707 + 0.707i)12-s + (0.956 − 0.290i)13-s + (−0.555 − 0.831i)14-s + (0.881 + 0.471i)15-s + (0.707 + 0.707i)16-s + (−0.0980 − 0.995i)17-s + ⋯
L(s,χ)  = 1  + (0.980 + 0.195i)2-s + (−0.382 + 0.923i)3-s + (0.923 + 0.382i)4-s + (0.0980 − 0.995i)5-s + (−0.555 + 0.831i)6-s + (−0.707 − 0.707i)7-s + (0.831 + 0.555i)8-s + (−0.707 − 0.707i)9-s + (0.290 − 0.956i)10-s + (0.773 + 0.634i)11-s + (−0.707 + 0.707i)12-s + (0.956 − 0.290i)13-s + (−0.555 − 0.831i)14-s + (0.881 + 0.471i)15-s + (0.707 + 0.707i)16-s + (−0.0980 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.981 + 0.189i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.981 + 0.189i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(193\)
\( \varepsilon \)  =  $0.981 + 0.189i$
motivic weight  =  \(0\)
character  :  $\chi_{193} (89, \cdot )$
Sato-Tate  :  $\mu(64)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 193,\ (1:\ ),\ 0.981 + 0.189i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.218879395 + 0.3072965172i$
$L(\frac12,\chi)$  $\approx$  $3.218879395 + 0.3072965172i$
$L(\chi,1)$  $\approx$  1.840023874 + 0.2691647042i
$L(1,\chi)$  $\approx$  1.840023874 + 0.2691647042i

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

−26.564359396744785558307619456398, −25.340935044888976244697637362083, −24.92587314546563797634553568181, −23.67363940619561792866597058535, −22.9649602130353006495497779858, −22.13178413483733720786364480152, −21.5359214342877858032646724958, −19.907059713497504323014876702760, −19.03968570629560416093441274859, −18.50879174231360497740802466037, −17.0552399936252177047507327439, −15.91510383757139523728914444116, −14.86037000197162786807805049588, −13.796486493498377217281387857191, −13.16544084893218601768405955936, −11.89410529745309047659870676754, −11.36413836556265113996475773080, −10.208406774509999693857426831983, −8.49512793516595718016796330630, −6.8652103426993257694117681354, −6.39014601070554412783016301766, −5.51710788201127223948845769056, −3.599851229186005296744073477551, −2.69099927311506980787215070452, −1.31802605830122143489793123402, 1.01735229399589752634328508263, 3.2099371891032016905584092906, 4.16095638928641645916552526222, 5.05259627097109443229610456671, 6.099322070835111950254588342428, 7.28641925068895349252532188267, 8.93913984034238030966921347704, 9.91291125293583206872219157081, 11.18063976499849374805306874821, 12.09286277885253497734254484263, 13.156135970948121342892492548639, 14.04145256045397070090535443187, 15.31180967214085177999420599043, 16.198649190077845351358709891783, 16.66740228456472378321198743776, 17.706501612059468028636201722706, 19.82839814971020914416075908778, 20.41574614007772362206287071096, 21.07124302161231908182527937559, 22.337851931421214596084501946749, 22.889145847660279178345315116873, 23.696026989305563835639469195038, 24.9778505969482664395638669302, 25.63818152447427182569778391099, 26.81805125055234578115444872933

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