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Properties

Label 1-71-71.3-r0-0-0
Degree $1$
Conductor $71$
Sign $-0.998 - 0.0614i$
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  + (0.134 + 0.990i)2-s + (−0.550 − 0.834i)3-s + (−0.963 + 0.266i)4-s + (−0.809 + 0.587i)5-s + (0.753 − 0.657i)6-s + (−0.691 + 0.722i)7-s + (−0.393 − 0.919i)8-s + (−0.393 + 0.919i)9-s + (−0.691 − 0.722i)10-s + (−0.995 − 0.0896i)11-s + (0.753 + 0.657i)12-s + (−0.995 + 0.0896i)13-s + (−0.809 − 0.587i)14-s + (0.936 + 0.351i)15-s + (0.858 − 0.512i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (0.134 + 0.990i)2-s + (−0.550 − 0.834i)3-s + (−0.963 + 0.266i)4-s + (−0.809 + 0.587i)5-s + (0.753 − 0.657i)6-s + (−0.691 + 0.722i)7-s + (−0.393 − 0.919i)8-s + (−0.393 + 0.919i)9-s + (−0.691 − 0.722i)10-s + (−0.995 − 0.0896i)11-s + (0.753 + 0.657i)12-s + (−0.995 + 0.0896i)13-s + (−0.809 − 0.587i)14-s + (0.936 + 0.351i)15-s + (0.858 − 0.512i)16-s + (0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.998 - 0.0614i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ -0.998 - 0.0614i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008491461097 + 0.2759367663i\)
\(L(\frac12)\) \(\approx\) \(0.008491461097 + 0.2759367663i\)
\(L(1)\) \(\approx\) \(0.4291529565 + 0.2873547158i\)
\(L(1)\) \(\approx\) \(0.4291529565 + 0.2873547158i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.134 + 0.990i)T \)
3 \( 1 + (-0.550 - 0.834i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.691 + 0.722i)T \)
11 \( 1 + (-0.995 - 0.0896i)T \)
13 \( 1 + (-0.995 + 0.0896i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.936 - 0.351i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-0.0448 + 0.998i)T \)
31 \( 1 + (0.858 + 0.512i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (0.473 - 0.880i)T \)
47 \( 1 + (-0.550 + 0.834i)T \)
53 \( 1 + (-0.963 - 0.266i)T \)
59 \( 1 + (0.753 + 0.657i)T \)
61 \( 1 + (-0.691 - 0.722i)T \)
67 \( 1 + (-0.963 + 0.266i)T \)
73 \( 1 + (0.134 + 0.990i)T \)
79 \( 1 + (-0.393 - 0.919i)T \)
83 \( 1 + (0.753 + 0.657i)T \)
89 \( 1 + (-0.963 - 0.266i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−31.499788606305134835728523516200, −29.75755922579475365427360978049, −28.92070637951300771145086746604, −28.05648030039012528394053658126, −26.98452904640747474161479213902, −26.37825141852278295744031641272, −24.105937635606786284198643020179, −23.01963171750021747049867571804, −22.462988081239169851158524235346, −20.95801362686040666446526974589, −20.29857672627368689624711635269, −19.22275729563063604909547344036, −17.70901837551388970356797752669, −16.47337139084398181569330760365, −15.46954020518875210431298625861, −13.822935206725208228724375109135, −12.42604275053871375093326580146, −11.595136924649488047773037131056, −10.22792173997149533119256683489, −9.5070278467813976921723828042, −7.72514658810739839275565661843, −5.39745410591807751135533706819, −4.35492178234084544837278616608, −3.15558612100620440940967396263, −0.32178154833034830476105345051, 2.98788271166085887148448147673, 5.039750927886499711479134343042, 6.28842297157690522563675221763, 7.35085638166833273349472560407, 8.33109536802639074042623389610, 10.218977197777907768098299233863, 12.00682520951696089825061287647, 12.78975123218302581971007929945, 14.21179305932152680201959029044, 15.49782937985420702473707838510, 16.36290991353125768441600836597, 17.78677574336538086948009853349, 18.6737425348688368068411867132, 19.494826155110234818801676400340, 21.92881111153974459130123721868, 22.575273238634362313823355704814, 23.68547054327314600726533362190, 24.35465449794328554894856419103, 25.66916709462295543586953893120, 26.536848974661099551434170216010, 27.89390412447425495650003535803, 28.91307970551790025851937307317, 30.376593873477367222022009729465, 31.26681278219021498125863770892, 32.11814778403276082339621191153

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