↑

Properties

Label 1-71-71.18-r0-0-0
Degree $1$
Conductor $71$
Sign $0.927 - 0.373i$
Analytic cond. $0.329722$
Root an. cond. $0.329722$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.983 − 0.178i)2-s + (−0.963 − 0.266i)3-s + (0.936 − 0.351i)4-s + (0.309 + 0.951i)5-s + (−0.995 − 0.0896i)6-s + (0.473 − 0.880i)7-s + (0.858 − 0.512i)8-s + (0.858 + 0.512i)9-s + (0.473 + 0.880i)10-s + (−0.393 − 0.919i)11-s + (−0.995 + 0.0896i)12-s + (−0.393 + 0.919i)13-s + (0.309 − 0.951i)14-s + (−0.0448 − 0.998i)15-s + (0.753 − 0.657i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.983 − 0.178i)2-s + (−0.963 − 0.266i)3-s + (0.936 − 0.351i)4-s + (0.309 + 0.951i)5-s + (−0.995 − 0.0896i)6-s + (0.473 − 0.880i)7-s + (0.858 − 0.512i)8-s + (0.858 + 0.512i)9-s + (0.473 + 0.880i)10-s + (−0.393 − 0.919i)11-s + (−0.995 + 0.0896i)12-s + (−0.393 + 0.919i)13-s + (0.309 − 0.951i)14-s + (−0.0448 − 0.998i)15-s + (0.753 − 0.657i)16-s + (−0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ 0.927 - 0.373i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.299016150 - 0.2517696480i\)
\(L(\frac12)\) \(\approx\) \(1.299016150 - 0.2517696480i\)
\(L(1)\) \(\approx\) \(1.384812110 - 0.1942767173i\)
\(L(1)\) \(\approx\) \(1.384812110 - 0.1942767173i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.983 - 0.178i)T \)
3 \( 1 + (-0.963 - 0.266i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.473 - 0.880i)T \)
11 \( 1 + (-0.393 - 0.919i)T \)
13 \( 1 + (-0.393 + 0.919i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.0448 + 0.998i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (-0.550 + 0.834i)T \)
31 \( 1 + (0.753 + 0.657i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (0.134 - 0.990i)T \)
47 \( 1 + (-0.963 + 0.266i)T \)
53 \( 1 + (0.936 + 0.351i)T \)
59 \( 1 + (-0.995 + 0.0896i)T \)
61 \( 1 + (0.473 + 0.880i)T \)
67 \( 1 + (0.936 - 0.351i)T \)
73 \( 1 + (0.983 - 0.178i)T \)
79 \( 1 + (0.858 - 0.512i)T \)
83 \( 1 + (-0.995 + 0.0896i)T \)
89 \( 1 + (0.936 + 0.351i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.99867938640643768494647824226, −30.78127972370282780916928421286, −29.63469528451597796427674567910, −28.3539436233030331422222461070, −28.06524845429164058918467406419, −26.1097439102413811061325083117, −24.646661829883123656154716398506, −24.20295780528453481469548301837, −22.87973552096459356719900636475, −21.93721328294408718557217793777, −21.06858489162267105066558361, −20.03365181219045782864286915593, −17.8701123681644336521546054742, −17.12555690145282704229867903533, −15.70440758881739074592784208869, −15.1191861535470717493150895663, −13.150611225346388820270765232750, −12.42063887449582962753810795985, −11.40320866110739972181846993919, −9.91344631311673757766113345985, −8.07943402509117714086875522159, −6.325251660229186407086374835859, −5.20221513230473308839827611192, −4.49731145791627097785928882474, −2.119240940827947921976909676745, 1.90329540841029499622104594657, 3.81599403029472994333496639911, 5.27466563680307196031043878117, 6.517326552927753669745014810892, 7.41474384758574824699021352775, 10.270576997318244301936202854303, 11.01821145475047528633035291435, 11.98054535891159475670080046198, 13.59540924809674154320138062688, 14.18819029838392230587227405341, 15.82981550597681065862210946270, 16.912193341533353951861292812470, 18.25864901783402869751291202443, 19.388455174452619462023086832445, 21.01676952233892365417126190915, 21.91213175277150165853440937740, 22.81692988240989747642782384503, 23.78281005392177376315186216925, 24.54592467165140025066302196455, 26.21312748345768049549058281681, 27.34663593678216276546729498625, 29.07550520563912167838387875295, 29.44684027451628429449294366710, 30.38797024410272454481880713086, 31.45890475252659583220335952391

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