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Properties

Label 1-71-71.11-r1-0-0
Degree $1$
Conductor $71$
Sign $0.305 - 0.952i$
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  + (−0.550 − 0.834i)2-s + (−0.995 − 0.0896i)3-s + (−0.393 + 0.919i)4-s + (−0.809 + 0.587i)5-s + (0.473 + 0.880i)6-s + (−0.936 + 0.351i)7-s + (0.983 − 0.178i)8-s + (0.983 + 0.178i)9-s + (0.936 + 0.351i)10-s + (−0.134 − 0.990i)11-s + (0.473 − 0.880i)12-s + (−0.134 + 0.990i)13-s + (0.809 + 0.587i)14-s + (0.858 − 0.512i)15-s + (−0.691 − 0.722i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.550 − 0.834i)2-s + (−0.995 − 0.0896i)3-s + (−0.393 + 0.919i)4-s + (−0.809 + 0.587i)5-s + (0.473 + 0.880i)6-s + (−0.936 + 0.351i)7-s + (0.983 − 0.178i)8-s + (0.983 + 0.178i)9-s + (0.936 + 0.351i)10-s + (−0.134 − 0.990i)11-s + (0.473 − 0.880i)12-s + (−0.134 + 0.990i)13-s + (0.809 + 0.587i)14-s + (0.858 − 0.512i)15-s + (−0.691 − 0.722i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ 0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3850946860 - 0.2808101693i\)
\(L(\frac12)\) \(\approx\) \(0.3850946860 - 0.2808101693i\)
\(L(1)\) \(\approx\) \(0.4468906925 - 0.1391167459i\)
\(L(1)\) \(\approx\) \(0.4468906925 - 0.1391167459i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.550 - 0.834i)T \)
3 \( 1 + (-0.995 - 0.0896i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.936 + 0.351i)T \)
11 \( 1 + (-0.134 - 0.990i)T \)
13 \( 1 + (-0.134 + 0.990i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.858 + 0.512i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (0.753 + 0.657i)T \)
31 \( 1 + (0.691 - 0.722i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
43 \( 1 + (-0.0448 + 0.998i)T \)
47 \( 1 + (0.995 - 0.0896i)T \)
53 \( 1 + (0.393 + 0.919i)T \)
59 \( 1 + (-0.473 + 0.880i)T \)
61 \( 1 + (-0.936 - 0.351i)T \)
67 \( 1 + (0.393 - 0.919i)T \)
73 \( 1 + (-0.550 - 0.834i)T \)
79 \( 1 + (0.983 - 0.178i)T \)
83 \( 1 + (0.473 - 0.880i)T \)
89 \( 1 + (-0.393 - 0.919i)T \)
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

−32.236024519031167231041378410010, −30.55665282814057923810057491869, −28.93533490377992248714174330072, −28.27741570342271890211977403382, −27.38645066335960588899665207880, −26.337739979530475803896367016499, −25.04665361181763797076822266300, −23.84312178148202880723718332291, −23.12937819373143743892663944808, −22.27864406609110478643928128588, −20.195476017229108111802647604172, −19.30935207543791670960169876333, −17.81632228204847470702675853231, −17.07272367231030451563875358242, −15.805984697644476248143575013020, −15.43236982886020068459502068453, −13.26395201852280385291142702055, −12.15969800512375474754184410277, −10.53877694459210002013684676405, −9.58318221916118896480144418377, −7.877140907023795600571421265963, −6.81554448422589226695377920157, −5.451995870816389264883942943669, −4.19856530539912718751079876118, −0.81591327600005352506437263316, 0.517470512321973052185417619539, 2.84045895379695491778880941945, 4.29641156832868864314998015326, 6.32084409924362161135445808933, 7.59933228906977612434567534968, 9.28704403962224496321041220164, 10.5649463648421784110748645350, 11.59641329976465778607683780070, 12.29642098530242532115957786953, 13.79081342220436572582869724841, 16.035323662670911639588146488655, 16.46368594060810419051979365633, 18.25315264460752010328480548889, 18.75908459361936768152761279250, 19.80176437163494307619230563795, 21.47510065927175768593684187161, 22.34243912993901772860209193873, 23.10812372698479320486852239708, 24.58235148156571506370177965787, 26.37232267932958553354034006581, 26.93585877355757581420613399148, 28.15788826795948470419303564814, 29.05555591327674777362294075897, 29.74292309455238494553081405550, 31.034241473774499551374719397921

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