Properties

Degree 1
Conductor 293
Sign $0.488 - 0.872i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.954 − 0.296i)2-s + (0.985 + 0.171i)3-s + (0.823 + 0.566i)4-s + (0.276 − 0.961i)5-s + (−0.890 − 0.455i)6-s + (−0.234 − 0.972i)7-s + (−0.618 − 0.785i)8-s + (0.941 + 0.337i)9-s + (−0.548 + 0.835i)10-s + (0.996 + 0.0859i)11-s + (0.714 + 0.699i)12-s + (0.584 + 0.811i)13-s + (−0.0645 + 0.997i)14-s + (0.436 − 0.899i)15-s + (0.357 + 0.933i)16-s + (−0.954 − 0.296i)17-s + ⋯
L(s,χ)  = 1  + (−0.954 − 0.296i)2-s + (0.985 + 0.171i)3-s + (0.823 + 0.566i)4-s + (0.276 − 0.961i)5-s + (−0.890 − 0.455i)6-s + (−0.234 − 0.972i)7-s + (−0.618 − 0.785i)8-s + (0.941 + 0.337i)9-s + (−0.548 + 0.835i)10-s + (0.996 + 0.0859i)11-s + (0.714 + 0.699i)12-s + (0.584 + 0.811i)13-s + (−0.0645 + 0.997i)14-s + (0.436 − 0.899i)15-s + (0.357 + 0.933i)16-s + (−0.954 − 0.296i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.488 - 0.872i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.488 - 0.872i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(293\)
\( \varepsilon \)  =  $0.488 - 0.872i$
motivic weight  =  \(0\)
character  :  $\chi_{293} (24, \cdot )$
Sato-Tate  :  $\mu(73)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 293,\ (0:\ ),\ 0.488 - 0.872i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.071134667 - 0.6282541042i$
$L(\frac12,\chi)$  $\approx$  $1.071134667 - 0.6282541042i$
$L(\chi,1)$  $\approx$  1.004133444 - 0.3202101131i
$L(1,\chi)$  $\approx$  1.004133444 - 0.3202101131i

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

−25.891552374645933191884929266001, −24.700946730689473863757232326259, −24.596007031068156694095338351217, −22.86056344159456134791396500162, −21.94047118706747667306572635139, −20.86254641607945666643321573622, −19.85324335369720867798030013508, −19.1975656223710522877605503409, −18.23206989282212381915881357462, −17.92798389927591366361214261927, −16.35426798831495435120014216416, −15.380170839443020910118322183743, −14.78167181313610914075288014072, −13.9523231557658556124213729998, −12.56965335502934992504991364041, −11.386715933260648849176383182453, −10.28223308618392109397813910229, −9.40618159691416074492423545850, −8.61040430191585869045056476726, −7.680943458495844579989615039303, −6.54070314717810668432595773137, −5.88155934411454497061947540477, −3.657868933673988267024294031103, −2.5874045676786511534052583532, −1.660507325653424401850562633834, 1.11889052072880023000312619545, 2.0982323115814526501650721199, 3.64713503289470717603421530938, 4.42453746447247363460713390817, 6.49275939901378316594132763079, 7.381989078019401604870505456022, 8.54826745337196119417360559954, 9.21526456720294871369261422117, 9.85260150814701911928318803278, 11.09142393563024077005671208614, 12.18984411877471285453164003822, 13.42226117914438586391740728309, 13.94737988648448007135523818180, 15.575063873199883019723442609159, 16.20956560364585248858326196323, 17.13939612040639954974478976236, 17.96010560630739209147435174209, 19.42489263384432463065543099707, 19.73713146147525940053359870058, 20.57226766085518994804398072823, 21.22387057084610326252086092070, 22.30246277860221475142610821536, 24.091776736909577685377369131537, 24.48791127041950619332453290584, 25.61526772553550495976273423852

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