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Properties

Label 1-73-73.40-r1-0-0
Degree $1$
Conductor $73$
Sign $0.306 + 0.951i$
Analytic cond. $7.84493$
Root an. cond. $7.84493$
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 + (0.866 + 0.5i)3-s + (−0.939 − 0.342i)4-s + (−0.573 + 0.819i)5-s + (0.642 − 0.766i)6-s + (−0.965 + 0.258i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (0.707 + 0.707i)10-s + (0.819 + 0.573i)11-s + (−0.642 − 0.766i)12-s + (−0.996 + 0.0871i)13-s + (0.0871 + 0.996i)14-s + (−0.906 + 0.422i)15-s + (0.766 + 0.642i)16-s + (−0.258 + 0.965i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.866 + 0.5i)3-s + (−0.939 − 0.342i)4-s + (−0.573 + 0.819i)5-s + (0.642 − 0.766i)6-s + (−0.965 + 0.258i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (0.707 + 0.707i)10-s + (0.819 + 0.573i)11-s + (−0.642 − 0.766i)12-s + (−0.996 + 0.0871i)13-s + (0.0871 + 0.996i)14-s + (−0.906 + 0.422i)15-s + (0.766 + 0.642i)16-s + (−0.258 + 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.306 + 0.951i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ 0.306 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.018425782 + 0.7417373369i\)
\(L(\frac12)\) \(\approx\) \(1.018425782 + 0.7417373369i\)
\(L(1)\) \(\approx\) \(1.053855763 + 0.05262985043i\)
\(L(1)\) \(\approx\) \(1.053855763 + 0.05262985043i\)

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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.573 + 0.819i)T \)
7 \( 1 + (-0.965 + 0.258i)T \)
11 \( 1 + (0.819 + 0.573i)T \)
13 \( 1 + (-0.996 + 0.0871i)T \)
17 \( 1 + (-0.258 + 0.965i)T \)
19 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (0.573 + 0.819i)T \)
31 \( 1 + (0.422 - 0.906i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.258 - 0.965i)T \)
47 \( 1 + (0.0871 - 0.996i)T \)
53 \( 1 + (-0.819 + 0.573i)T \)
59 \( 1 + (-0.0871 - 0.996i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (-0.642 + 0.766i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (0.642 - 0.766i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−31.60875466138675233177102378031, −30.24100900979572264348480573323, −29.03547552253841732195296269639, −27.27008587520759819256511016441, −26.702614566941194211086392742187, −25.23461774460834987083325512862, −24.77965021235887952512728430955, −23.67971196941805217918946944314, −22.66714924214774024366255458834, −21.17333590653261276490029562425, −19.628120680955905852415134757400, −19.12323244828661542522269512316, −17.389709424486275157942034273743, −16.35560059082870517630991868908, −15.306933905865385175194247300640, −14.09986754497979825599930769109, −13.04875509407421406867221678588, −12.178127332629370406780707356603, −9.52772439723326906551155055382, −8.734969184969668204113252654804, −7.48783781424689271237847834666, −6.45713693516702205543914145696, −4.56072734338215920854184320305, −3.2652728708267837707687532937, −0.53636920448869303135842691074, 2.310318407398320587138362236789, 3.41013232658248460650450660179, 4.49908614315020668466944319910, 6.75519038814665091980593455359, 8.53127104735048636275305810386, 9.71072174323837357605798223092, 10.58733058742408519884476797028, 12.0767239881532685684634007350, 13.21844174317497687292704144214, 14.697354348398368219382027549258, 15.17251466694397726713645036557, 17.06129207137279562284848557148, 18.88735740204825975673944927837, 19.40331547282929218617827376862, 20.22722669395314550850061202809, 21.8407401323169829655819021375, 22.21933200021484728448133522701, 23.51319207252403116068705758184, 25.27893539943058023280355481605, 26.3583846632106267050307587173, 27.193149052986666470966643298263, 28.182788415715288688113466614800, 29.59551637936887182412705201605, 30.519480954404581840827114655728, 31.38076280857478829913398385153

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