Properties

Degree 1
Conductor 193
Sign $-0.336 + 0.941i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.336 + 0.941i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.336 + 0.941i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(193\)
\( \varepsilon \)  =  $-0.336 + 0.941i$
motivic weight  =  \(0\)
character  :  $\chi_{193} (164, \cdot )$
Sato-Tate  :  $\mu(64)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 193,\ (1:\ ),\ -0.336 + 0.941i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.5435109522 - 0.7718104182i$
$L(\frac12,\chi)$  $\approx$  $-0.5435109522 - 0.7718104182i$
$L(\chi,1)$  $\approx$  0.5841910041 - 0.7948053179i
$L(1,\chi)$  $\approx$  0.5841910041 - 0.7948053179i

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

−27.41135158325732176071228397263, −26.078806236527931663543435056726, −25.62901760367068835765176256185, −24.95165780418670401086004811052, −23.758701528727003948443773591843, −22.645087212172930801764543035320, −21.79845355279559303618698682460, −21.09868356345510952578465677263, −19.96911414824461302656451909433, −18.66772233000170992642096557304, −17.434884394385935381046414564047, −16.652217748291649082066659278464, −15.68263368411148859236295419923, −15.17441487510013182720274337043, −13.80089890049224582103516979089, −13.16409768185798782705501969095, −11.88644167935546520756720904613, −10.0428376320114034096175633037, −9.10828738919634324514520750731, −8.81293483221606614179695861991, −7.11571468414267007169943557485, −5.84852654084053738334663563542, −4.916771680043656327425158661167, −3.97186566192984706343160481224, −2.32925645514180319673120335203, 0.27758290109487275252878505005, 1.686632011331462706917521445045, 2.93832019417416860664832610571, 3.69218559840666334354404915939, 5.79880192819521124269155566231, 6.55664463985247067331852427135, 8.09716023767826439634989746374, 9.162661366984756977334889393242, 10.526856627683666273224146888533, 11.00387989635926644059569061335, 12.736615441607396540567652970226, 13.12355477981472056794254586701, 14.08237387894724553004237052852, 14.871089530173058313435456737708, 16.78368919887972059599467460894, 17.90773862047928063802463570246, 18.61701013809744104367981409341, 19.4312418488011033032254061479, 20.24355183047915097129923805755, 21.31317280325733803068394202922, 22.381813091112310146089414366780, 23.12563143884382700653748625706, 24.0447975397969382370255431715, 25.32977812788148654657422893852, 26.23335139382795763677863900728

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