Properties

Degree 1
Conductor 193
Sign $-0.770 + 0.637i$
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 + (0.995 + 0.0980i)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 + (−0.956 − 0.290i)10-s + (0.634 − 0.773i)11-s + (−0.707 + 0.707i)12-s + (0.290 + 0.956i)13-s + (0.555 + 0.831i)14-s + (−0.471 + 0.881i)15-s + (0.707 + 0.707i)16-s + (−0.995 + 0.0980i)17-s + ⋯
L(s,χ)  = 1  + (−0.980 − 0.195i)2-s + (−0.382 + 0.923i)3-s + (0.923 + 0.382i)4-s + (0.995 + 0.0980i)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 + (−0.956 − 0.290i)10-s + (0.634 − 0.773i)11-s + (−0.707 + 0.707i)12-s + (0.290 + 0.956i)13-s + (0.555 + 0.831i)14-s + (−0.471 + 0.881i)15-s + (0.707 + 0.707i)16-s + (−0.995 + 0.0980i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.770 + 0.637i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.770 + 0.637i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(193\)
\( \varepsilon \)  =  $-0.770 + 0.637i$
motivic weight  =  \(0\)
character  :  $\chi_{193} (125, \cdot )$
Sato-Tate  :  $\mu(64)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 193,\ (1:\ ),\ -0.770 + 0.637i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2033822511 + 0.5645485137i$
$L(\frac12,\chi)$  $\approx$  $0.2033822511 + 0.5645485137i$
$L(\chi,1)$  $\approx$  0.5867373320 + 0.1789686842i
$L(1,\chi)$  $\approx$  0.5867373320 + 0.1789686842i

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

−26.05810120580421225732773830419, −25.428616694617820304143244842845, −24.8726935092598693037417040610, −23.9738725985243500467938183119, −22.60970545777688210789327014851, −21.81774836827095639621937286205, −20.20405036809422659680886696001, −19.6595260480361720694966147018, −18.36436751711261648799851417349, −17.853773570410740694067378039254, −17.15529735515429712677856510503, −15.992077824273923675213680233539, −14.90268402064492256107239318700, −13.48105768368260913558461408098, −12.5797817069488823915063196555, −11.53662051912043128404140512747, −10.3048277457741289562698457720, −9.30629526523327682162425157401, −8.38766242264321887703539122978, −6.93866838126156874236086363493, −6.30316227383307201349290762420, −5.34288642486381614973689922166, −2.65069611504646051670567548281, −1.794791282265032338497181997640, −0.3018492107949454590961095704, 1.36420831254100622073643655752, 3.03184544256131687156642200175, 4.221317461787987095540504876487, 6.19943619697577511034732440757, 6.512475832916919539302821591908, 8.47736733842600905897162956375, 9.421242784547125610742111596036, 10.12126886755636932594437373727, 10.95315072779265107229321013639, 11.99791926561601151187796327415, 13.525419057554691492378594988786, 14.59356869744884964196905768325, 16.13780495296486962345708211559, 16.53831390782570945900427324359, 17.38770182718181818186567993681, 18.367503305339702004642223238, 19.59690047173385157829259481199, 20.436310169556467247728286488872, 21.53083846871614517554345062881, 21.96386214217800054461775788774, 23.31136460431382598945384709477, 24.644311613644412343095226206882, 25.6510118872014876173628340068, 26.50973082672585445528052402547, 26.915376901899033353065801910578

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