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Properties

Label 1-293-293.289-r0-0-0
Degree $1$
Conductor $293$
Sign $0.940 + 0.338i$
Analytic cond. $1.36068$
Root an. cond. $1.36068$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $0.940 + 0.338i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ 0.940 + 0.338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5248280740 + 0.09153220236i\)
\(L(\frac12)\) \(\approx\) \(0.5248280740 + 0.09153220236i\)
\(L(1)\) \(\approx\) \(0.5456319300 - 0.04627245407i\)
\(L(1)\) \(\approx\) \(0.5456319300 - 0.04627245407i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (-0.999 - 0.0430i)T \)
3 \( 1 + (-0.890 - 0.455i)T \)
5 \( 1 + (-0.397 - 0.917i)T \)
7 \( 1 + (0.966 - 0.255i)T \)
11 \( 1 + (-0.234 + 0.972i)T \)
13 \( 1 + (0.512 + 0.858i)T \)
17 \( 1 + (-0.999 - 0.0430i)T \)
19 \( 1 + (-0.0645 + 0.997i)T \)
23 \( 1 + (0.869 - 0.493i)T \)
29 \( 1 + (-0.744 + 0.668i)T \)
31 \( 1 + (-0.397 + 0.917i)T \)
37 \( 1 + (0.941 + 0.337i)T \)
41 \( 1 + (0.714 + 0.699i)T \)
43 \( 1 + (-0.925 - 0.377i)T \)
47 \( 1 + (0.276 + 0.961i)T \)
53 \( 1 + (-0.234 + 0.972i)T \)
59 \( 1 + (0.436 - 0.899i)T \)
61 \( 1 + (0.823 - 0.566i)T \)
67 \( 1 + (-0.618 + 0.785i)T \)
71 \( 1 + (0.651 + 0.758i)T \)
73 \( 1 + (0.996 - 0.0859i)T \)
79 \( 1 + (0.823 - 0.566i)T \)
83 \( 1 + (-0.150 - 0.988i)T \)
89 \( 1 + (0.823 + 0.566i)T \)
97 \( 1 + (0.908 - 0.417i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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