Properties

Label 1-73-73.2-r0-0-0
Degree $1$
Conductor $73$
Sign $-0.194 - 0.980i$
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  + (0.766 − 0.642i)2-s + (−0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)6-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + 10-s + (0.766 + 0.642i)11-s + (−0.939 + 0.342i)12-s + (−0.939 − 0.342i)13-s + (−0.939 − 0.342i)14-s + (0.173 − 0.984i)15-s + (−0.939 − 0.342i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)6-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + 10-s + (0.766 + 0.642i)11-s + (−0.939 + 0.342i)12-s + (−0.939 − 0.342i)13-s + (−0.939 − 0.342i)14-s + (0.173 − 0.984i)15-s + (−0.939 − 0.342i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.194 - 0.980i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ -0.194 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7870351753 - 0.9587100501i\)
\(L(\frac12)\) \(\approx\) \(0.7870351753 - 0.9587100501i\)
\(L(1)\) \(\approx\) \(1.067595500 - 0.7692809331i\)
\(L(1)\) \(\approx\) \(1.067595500 - 0.7692809331i\)

Euler product

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

−32.0799140472923617432919492606, −31.32505971263098151796212269370, −29.41902871106197625708013455451, −28.89354279830233186261282902601, −27.38546814851673303347077098756, −26.36762205974479587194216348743, −25.00898868099110453727047432795, −24.472640647992298409621774027414, −22.83200182638407133294759682790, −21.993041050696727607972033561792, −21.386076776190404806948418584690, −20.15765886149020228275675331909, −18.100607630435774991134222386960, −16.77811422552402282279108921907, −16.32841480837927979786480238911, −15.02359351472443731723110151139, −13.9234716567792415904085485143, −12.472746559786627792519944916616, −11.58930802277074777433937964953, −9.61251127120873094582369183243, −8.77403742266912525147272623016, −6.56758196426818504564923803792, −5.569683982265199917163640197229, −4.59261606799748548642688590668, −2.89718301436746889267951258882, 1.52170726701077121631606929227, 3.00188476023841031011816118727, 4.8942945051503390342885588419, 6.390040695925734562791770819165, 7.10156078191428579224024508770, 9.69935154727712667292774121087, 10.68320734841135990119582733920, 11.9196905910505250189908175430, 13.11451802107246129859082813560, 13.84299698856010255197387260135, 15.053842158491011455355604051148, 17.03962700214038601524369584313, 17.90571105400558463325363947288, 19.36807520726128312719233247060, 19.96578948620641239168248564261, 21.725930698941975279892234730966, 22.505081336849408529383705838618, 23.30984529719498864364332953441, 24.52452294281426015733505121620, 25.49649581511352283741743047447, 27.08915553439617980914014976661, 28.695371157020855031206756677597, 29.20381673325601713006623324424, 30.29115860544085422548638148658, 30.64809364293444331610314361225

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