Properties

Degree 1
Conductor 73
Sign $0.641 + 0.766i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.173 − 0.984i)2-s + (−0.5 + 0.866i)3-s + (−0.939 − 0.342i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)6-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + 10-s + (0.173 + 0.984i)11-s + (0.766 − 0.642i)12-s + (0.766 + 0.642i)13-s + (0.766 + 0.642i)14-s + (−0.939 − 0.342i)15-s + (0.766 + 0.642i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  + (0.173 − 0.984i)2-s + (−0.5 + 0.866i)3-s + (−0.939 − 0.342i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)6-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + 10-s + (0.173 + 0.984i)11-s + (0.766 − 0.642i)12-s + (0.766 + 0.642i)13-s + (0.766 + 0.642i)14-s + (−0.939 − 0.342i)15-s + (0.766 + 0.642i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.641 + 0.766i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.641 + 0.766i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $0.641 + 0.766i$
motivic weight  =  \(0\)
character  :  $\chi_{73} (4, \cdot )$
Sato-Tate  :  $\mu(9)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 73,\ (0:\ ),\ 0.641 + 0.766i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6513959665 + 0.3043017687i$
$L(\frac12,\chi)$  $\approx$  $0.6513959665 + 0.3043017687i$
$L(\chi,1)$  $\approx$  0.8260485188 + 0.09446651387i
$L(1,\chi)$  $\approx$  0.8260485188 + 0.09446651387i

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

−31.60169255680601175787496415725, −30.36215276100827343519675428656, −29.27479507053047594264994004971, −28.16931579277892121225038391251, −26.97298206257873353007258719487, −25.627144783595851332408299398310, −24.71003262383711095438637267882, −23.854285801652130051186821471740, −23.0896662547019101433894508567, −21.90656638176436730221114913697, −20.2726057827804756238301485091, −18.93307460074153997536981348370, −17.72353066638377311169866923558, −16.686784880035827799677597327926, −16.18434701728046959997989523012, −14.19333058864295489456942552481, −13.17831011154087599107577263805, −12.57212292318081332353040628396, −10.67269426653254764098497121955, −8.807548596680715413378457406652, −7.84600732197490033821538866291, −6.39937698872839342152807707305, −5.544172147258125939649737185822, −3.875906189365897787527532331908, −0.89830344698635881097842190160, 2.43896404418950992236546620670, 3.74441913688632424837962867206, 5.19863689264433947588783077453, 6.579153975152450081550745272470, 9.14342914084500281575452405174, 9.815080047290832286946547141360, 11.16372465095747407322881149361, 11.840131247270514284363612090506, 13.4460284093829038627560741846, 14.83134633521739624169753215598, 15.730100757409793335262572091063, 17.59430499632110237980766682733, 18.36652569871775522289379403155, 19.6397601762360163145693668081, 20.9532126941604795429492529462, 21.912123814894314204017323986793, 22.54328855617124048526802190039, 23.44244984207594345764840180982, 25.58532680484202138853797740488, 26.51418945956717741180770391036, 27.66998468872257619135826551058, 28.52657090192076546056774819479, 29.32887744727761061134364504244, 30.69671865593801675656244810864, 31.473639521092433956745302026

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