Properties

Degree 1
Conductor 73
Sign $-0.194 - 0.980i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.766 − 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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $-0.194 - 0.980i$
motivic weight  =  \(0\)
character  :  $\chi_{73} (2, \cdot )$
Sato-Tate  :  $\mu(9)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 73,\ (0:\ ),\ -0.194 - 0.980i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7870351753 - 0.9587100501i$
$L(\frac12,\chi)$  $\approx$  $0.7870351753 - 0.9587100501i$
$L(\chi,1)$  $\approx$  1.067595500 - 0.7692809331i
$L(1,\chi)$  $\approx$  1.067595500 - 0.7692809331i

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

−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