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Properties

Label 1-73-73.55-r0-0-0
Degree $1$
Conductor $73$
Sign $0.641 - 0.766i$
Analytic cond. $0.339010$
Root an. cond. $0.339010$
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  + (0.173 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.641 - 0.766i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ 0.641 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6513959665 - 0.3043017687i\)
\(L(\frac12)\) \(\approx\) \(0.6513959665 - 0.3043017687i\)
\(L(1)\) \(\approx\) \(0.8260485188 - 0.09446651387i\)
\(L(1)\) \(\approx\) \(0.8260485188 - 0.09446651387i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.173 - 0.984i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

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

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