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Properties

Label 1-71-71.8-r0-0-0
Degree $1$
Conductor $71$
Sign $0.999 + 0.0297i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.963 − 0.266i)2-s + (−0.393 − 0.919i)3-s + (0.858 + 0.512i)4-s + (0.309 + 0.951i)5-s + (0.134 + 0.990i)6-s + (−0.0448 + 0.998i)7-s + (−0.691 − 0.722i)8-s + (−0.691 + 0.722i)9-s + (−0.0448 − 0.998i)10-s + (0.983 − 0.178i)11-s + (0.134 − 0.990i)12-s + (0.983 + 0.178i)13-s + (0.309 − 0.951i)14-s + (0.753 − 0.657i)15-s + (0.473 + 0.880i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−0.963 − 0.266i)2-s + (−0.393 − 0.919i)3-s + (0.858 + 0.512i)4-s + (0.309 + 0.951i)5-s + (0.134 + 0.990i)6-s + (−0.0448 + 0.998i)7-s + (−0.691 − 0.722i)8-s + (−0.691 + 0.722i)9-s + (−0.0448 − 0.998i)10-s + (0.983 − 0.178i)11-s + (0.134 − 0.990i)12-s + (0.983 + 0.178i)13-s + (0.309 − 0.951i)14-s + (0.753 − 0.657i)15-s + (0.473 + 0.880i)16-s + (−0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.999 + 0.0297i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ 0.999 + 0.0297i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5957314450 + 0.008865830676i\)
\(L(\frac12)\) \(\approx\) \(0.5957314450 + 0.008865830676i\)
\(L(1)\) \(\approx\) \(0.6736463254 - 0.04834246530i\)
\(L(1)\) \(\approx\) \(0.6736463254 - 0.04834246530i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.963 - 0.266i)T \)
3 \( 1 + (-0.393 - 0.919i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.0448 + 0.998i)T \)
11 \( 1 + (0.983 - 0.178i)T \)
13 \( 1 + (0.983 + 0.178i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.753 + 0.657i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-0.995 + 0.0896i)T \)
31 \( 1 + (0.473 - 0.880i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-0.550 + 0.834i)T \)
47 \( 1 + (-0.393 + 0.919i)T \)
53 \( 1 + (0.858 - 0.512i)T \)
59 \( 1 + (0.134 - 0.990i)T \)
61 \( 1 + (-0.0448 - 0.998i)T \)
67 \( 1 + (0.858 + 0.512i)T \)
73 \( 1 + (-0.963 - 0.266i)T \)
79 \( 1 + (-0.691 - 0.722i)T \)
83 \( 1 + (0.134 - 0.990i)T \)
89 \( 1 + (0.858 - 0.512i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−32.31829948055808001846513742621, −30.33555240212833185674675416672, −29.02917894759565865752082170269, −28.2776716144546748568160888054, −27.40319420953711259582435110099, −26.4513357104061833804958389720, −25.408555980047718642146900827614, −24.15691961196667036476695234021, −23.08770579252907444190003245774, −21.49796370033787437481258190344, −20.34544532399718398344098037324, −19.81052093938176509580121689163, −17.7695326539870167947538550858, −17.06917137362475189474870154911, −16.25628055605553038504141396728, −15.17731453587656036647688906546, −13.57731224950030059528231112983, −11.689341674944650078512908208752, −10.63993175928678449594479509122, −9.45945602198385474546289705260, −8.63963482095783595435650242250, −6.82283630099376182965117279452, −5.44607732997191855977816778796, −3.88790094876280927432721093723, −1.171710994250717458721178372289, 1.68748306588473742973661790719, 3.00557751154363191432795968566, 6.06222233221769581963336860650, 6.78659335024945062977513841727, 8.26328165089347551536872660036, 9.507212454413305602713983472332, 11.19251349299590554949416848599, 11.71276810249954275499506229347, 13.235755124446927319870193071283, 14.77254697584552982939863150210, 16.272351753765809741211990608074, 17.5310638367492508012275722593, 18.57854613136318857481242826418, 18.83219390646415896476819808835, 20.34345234953528497566391503739, 21.89263212473812541144438454020, 22.719777492336650826689038014487, 24.55398746233085829542169177770, 25.142282560013712812827061924408, 26.22452117483279187303382363562, 27.520164685264358429810328634501, 28.60150996546770436187013330647, 29.38251206625021369592123359899, 30.42769472470317150290726817520, 31.0905870187211255387556082165

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