Properties

Label 1-73-73.8-r0-0-0
Degree $1$
Conductor $73$
Sign $0.927 - 0.373i$
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  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + 7-s + 8-s + 9-s + 10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + 7-s + 8-s + 9-s + 10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ 0.927 - 0.373i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9410416867 - 0.1824016528i\)
\(L(\frac12)\) \(\approx\) \(0.9410416867 - 0.1824016528i\)
\(L(1)\) \(\approx\) \(1.007438692 - 0.1993754872i\)
\(L(1)\) \(\approx\) \(1.007438692 - 0.1993754872i\)

Euler product

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

−31.75763361427223077483613646744, −31.03663066977544343650550104839, −29.35981542609931439095372345416, −27.875809308446503237819739950901, −27.152368275994982443104124864920, −26.262455300384007488585696822612, −24.9808064378093677868592759805, −24.21285121942538884597019088393, −23.55935337006795795017268314164, −21.4561363789269550684514257228, −20.46202328816979213717894676632, −19.240476931000344304768758917366, −18.43347361306306250176439361733, −16.80523466507783422931016324695, −16.00632723097284044591235070469, −14.65541257912965393705691196270, −13.990530248514427279043360432886, −12.4135206552701795225515779001, −10.55088199714902184236459744746, −9.011646037015392759168497725, −8.28294768707544917572032769288, −7.3883683645836790468019409348, −5.35327045338311684881736376023, −4.08553503783596115196254133370, −1.60642668670275290712781566834, 2.01668485661567767382034843840, 3.17766410448070010016995199459, 4.577370201522971848124573512, 7.57097805563327699201604114370, 7.92906945612528956194629492225, 9.625831656541843497849012177422, 10.60839548784293371459828935510, 11.89419632608690371449419062097, 13.21258377790860175032007162889, 14.560117871655700864455908899779, 15.43453553254422151351743447951, 17.47684029479902530421887497852, 18.3881436022799132407780897818, 19.3899880735064161850010639041, 20.35500109715120011705491412705, 21.239305605646378376944332655559, 22.432841731337003076643078146322, 23.82936169736425687330353899379, 25.46846712415782882622518601743, 26.17853168183116024312716775420, 27.32813582931822567816888149137, 27.85257746555381096560850328700, 29.789631235959694182854531172043, 30.34051292645008162247575215710, 31.15553797638683711770299687269

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