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Properties

Label 1-71-71.68-r1-0-0
Degree $1$
Conductor $71$
Sign $0.984 + 0.177i$
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  + (0.134 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.984 + 0.177i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ 0.984 + 0.177i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217654794 + 0.1086522074i\)
\(L(\frac12)\) \(\approx\) \(1.217654794 + 0.1086522074i\)
\(L(1)\) \(\approx\) \(0.9049996153 + 0.1783540197i\)
\(L(1)\) \(\approx\) \(0.9049996153 + 0.1783540197i\)

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

−31.23092371063397350726974067251, −30.5286561466250319305326517732, −28.87957886430573734835570749003, −28.09386526347892502614112122944, −27.51500695195713228554264729041, −26.54894919362342778265202363037, −24.55541792939365312879239226284, −23.355516797227834458377448424022, −22.43373559241947626832029414678, −21.319223944242521191775191594807, −20.56787431717472872019992854613, −19.385030233351610554292849550122, −18.06214632409420211305285015003, −16.86849551368982495136990158987, −15.504834270627496057978904010638, −14.42583505357120029742767311946, −12.60293530430535518395602544664, −11.614385308763179979196127849154, −10.96438822511376269447889260399, −9.26773472116803046794980231064, −8.47394959602322391521192118896, −5.799042975552800207822854403284, −4.543882392095995809399336831175, −3.56347983055468334934037608973, −1.19025338790620409871764305635, 0.8524704615293112059496880398, 3.72910626477104430756874277591, 5.24612625955860943998903475678, 6.89682309099787987472220976661, 7.35721286752081142094132891593, 8.7591648017823158800542709754, 10.92553283325441227071491458902, 11.92997963537304611257183494144, 13.51712643910015095098041888098, 14.33074072122911904486733854392, 15.73597978968071452215197904533, 16.89898231192402233855798939515, 17.93262588047056006396663343015, 18.77520348242256456219055480728, 20.170254173683862122244070934248, 22.19640850550762825291419227596, 22.97134717753817033817001849006, 23.784629136708736510158151143506, 24.66447958773378967608044894610, 25.86194073315837210250684213819, 27.18603596266799724002601464684, 27.77623296021685261505937340473, 29.575459191415794939797474695716, 30.722524168764310520341327620747, 31.090972044781195055855875413077

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