Properties

Label 1-71-71.31-r1-0-0
Degree $1$
Conductor $71$
Sign $0.962 + 0.269i$
Analytic cond. $7.63000$
Root an. cond. $7.63000$
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  + (0.936 − 0.351i)2-s + (0.858 + 0.512i)3-s + (0.753 − 0.657i)4-s + (−0.809 + 0.587i)5-s + (0.983 + 0.178i)6-s + (0.550 + 0.834i)7-s + (0.473 − 0.880i)8-s + (0.473 + 0.880i)9-s + (−0.550 + 0.834i)10-s + (0.691 − 0.722i)11-s + (0.983 − 0.178i)12-s + (0.691 + 0.722i)13-s + (0.809 + 0.587i)14-s + (−0.995 + 0.0896i)15-s + (0.134 − 0.990i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.936 − 0.351i)2-s + (0.858 + 0.512i)3-s + (0.753 − 0.657i)4-s + (−0.809 + 0.587i)5-s + (0.983 + 0.178i)6-s + (0.550 + 0.834i)7-s + (0.473 − 0.880i)8-s + (0.473 + 0.880i)9-s + (−0.550 + 0.834i)10-s + (0.691 − 0.722i)11-s + (0.983 − 0.178i)12-s + (0.691 + 0.722i)13-s + (0.809 + 0.587i)14-s + (−0.995 + 0.0896i)15-s + (0.134 − 0.990i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.962 + 0.269i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ 0.962 + 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.466717883 + 0.4763155633i\)
\(L(\frac12)\) \(\approx\) \(3.466717883 + 0.4763155633i\)
\(L(1)\) \(\approx\) \(2.259882905 + 0.1344167059i\)
\(L(1)\) \(\approx\) \(2.259882905 + 0.1344167059i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.936 - 0.351i)T \)
3 \( 1 + (0.858 + 0.512i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.550 + 0.834i)T \)
11 \( 1 + (0.691 - 0.722i)T \)
13 \( 1 + (0.691 + 0.722i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.995 - 0.0896i)T \)
23 \( 1 + (-0.623 + 0.781i)T \)
29 \( 1 + (-0.393 - 0.919i)T \)
31 \( 1 + (-0.134 - 0.990i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (-0.963 - 0.266i)T \)
47 \( 1 + (-0.858 + 0.512i)T \)
53 \( 1 + (-0.753 - 0.657i)T \)
59 \( 1 + (-0.983 + 0.178i)T \)
61 \( 1 + (0.550 - 0.834i)T \)
67 \( 1 + (-0.753 + 0.657i)T \)
73 \( 1 + (0.936 - 0.351i)T \)
79 \( 1 + (0.473 - 0.880i)T \)
83 \( 1 + (0.983 - 0.178i)T \)
89 \( 1 + (0.753 + 0.657i)T \)
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

−31.31383359252913394360658727219, −30.44265393363225991712701615584, −29.9432490085042833290169301947, −28.115081826960644752632059201433, −26.77325285603133240673969302881, −25.635168585321302583834165737162, −24.64497063471837644121146791227, −23.72658400961636263992633846925, −23.031915566760337642937796607144, −21.273769363132224375478211284510, −20.16874748148274586946493994278, −19.785667413193923866476877786098, −17.795983317210931513739428536667, −16.52529082391551219466976175303, −15.14707952198676432625194750458, −14.43505284192501019301533567325, −13.059410399304767033501912746677, −12.37108132227966970938151177643, −10.84684523745775146230565493584, −8.56848422329686752773095499259, −7.749795802374767468513527203463, −6.54634571885125458994425375706, −4.47033511642088170744149708694, −3.63162981650125332484265732179, −1.59891918859142771598463128202, 2.152198369854516260379559238582, 3.49291083914676209823952845122, 4.51672563492048291980426000498, 6.299638987905728471430003501553, 7.951983054988892215925638280796, 9.353050407976871243931569856067, 11.08239019559945130636614797103, 11.70945645370171841860498279278, 13.49751470052786150404581612334, 14.4888950403225726804161389773, 15.308487162148763695936412102421, 16.221066388148419568611683408607, 18.70201720333147808153315889420, 19.36537174186391633013275746756, 20.5987797063561073489269545765, 21.59165255494037060488591091058, 22.3790526303522089173841687370, 23.7951647770434646803168949293, 24.77350664717832847378609685400, 25.92571312388168244723911743411, 27.31143830267681353017942314377, 28.058039790198491231285451820575, 29.78698595713755464095140857489, 30.69670558401770970938647661834, 31.480051184271159056500761094957

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