Properties

Degree 1
Conductor 293
Sign $0.940 - 0.338i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.999 + 0.0430i)2-s + (−0.890 + 0.455i)3-s + (0.996 − 0.0859i)4-s + (−0.397 + 0.917i)5-s + (0.869 − 0.493i)6-s + (0.966 + 0.255i)7-s + (−0.991 + 0.128i)8-s + (0.584 − 0.811i)9-s + (0.357 − 0.933i)10-s + (−0.234 − 0.972i)11-s + (−0.847 + 0.530i)12-s + (0.512 − 0.858i)13-s + (−0.976 − 0.213i)14-s + (−0.0645 − 0.997i)15-s + (0.985 − 0.171i)16-s + (−0.999 + 0.0430i)17-s + ⋯
L(s,χ)  = 1  + (−0.999 + 0.0430i)2-s + (−0.890 + 0.455i)3-s + (0.996 − 0.0859i)4-s + (−0.397 + 0.917i)5-s + (0.869 − 0.493i)6-s + (0.966 + 0.255i)7-s + (−0.991 + 0.128i)8-s + (0.584 − 0.811i)9-s + (0.357 − 0.933i)10-s + (−0.234 − 0.972i)11-s + (−0.847 + 0.530i)12-s + (0.512 − 0.858i)13-s + (−0.976 − 0.213i)14-s + (−0.0645 − 0.997i)15-s + (0.985 − 0.171i)16-s + (−0.999 + 0.0430i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.940 - 0.338i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.940 - 0.338i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(293\)
\( \varepsilon \)  =  $0.940 - 0.338i$
motivic weight  =  \(0\)
character  :  $\chi_{293} (73, \cdot )$
Sato-Tate  :  $\mu(73)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 293,\ (0:\ ),\ 0.940 - 0.338i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5248280740 - 0.09153220236i$
$L(\frac12,\chi)$  $\approx$  $0.5248280740 - 0.09153220236i$
$L(\chi,1)$  $\approx$  0.5456319300 + 0.04627245407i
$L(1,\chi)$  $\approx$  0.5456319300 + 0.04627245407i

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

−25.30584152447363276810016667036, −24.63905507344645651086173295552, −23.7839025669651462736559850149, −23.24850223795939308977643657748, −21.686943851720540199097908559, −20.71116554302058235694689405926, −20.114795486417346447220328107800, −18.89814784513454511200345872904, −18.13176713483885282649417743084, −17.329945609955715602738499968791, −16.62620283660601098188361230083, −15.860835079661055515766503135797, −14.67750123371155165383753315382, −13.06120921901406452505874240072, −12.24055520671717500397686593839, −11.3593710938552124325433527146, −10.71624611895624481476161959585, −9.37738039138718611467274509383, −8.34138861492322664534915971230, −7.48900844099762235826747030425, −6.57885812355725700404848897069, −5.20215589530198778557206740486, −4.24734624263047283770612257917, −1.94735755573816396233042611001, −1.21225201904438188567090950037, 0.63082810155467755533848394348, 2.41031429665735096798684486654, 3.730601302403050462453814497359, 5.33850371251396994660792927285, 6.25503001894999335363758470252, 7.317761667193483511014101064143, 8.3247815955377242228662595722, 9.41116428267227975157906707864, 10.81160429197704683713022660138, 11.035198233943470377210056331055, 11.686385949023110465483724070125, 13.27452306241325011313933471610, 15.07109928290682944968252377779, 15.26665106693294365813260549726, 16.344251038735958657418886640214, 17.44111761997372816216249531121, 18.04250609578390797152316804752, 18.71483573610509845460055202479, 19.83696939508460785785091410285, 20.98216591070238669886514284899, 21.68813662640774889023082697624, 22.66350211416070190439994002124, 23.82138356560591347891163735248, 24.37020000081492708452758689310, 25.66991486050572850651127374123

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