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Properties

Label 1-71-71.43-r0-0-0
Degree $1$
Conductor $71$
Sign $0.845 + 0.534i$
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  + (0.753 + 0.657i)2-s + (0.473 − 0.880i)3-s + (0.134 + 0.990i)4-s + (0.309 + 0.951i)5-s + (0.936 − 0.351i)6-s + (−0.393 − 0.919i)7-s + (−0.550 + 0.834i)8-s + (−0.550 − 0.834i)9-s + (−0.393 + 0.919i)10-s + (−0.0448 + 0.998i)11-s + (0.936 + 0.351i)12-s + (−0.0448 − 0.998i)13-s + (0.309 − 0.951i)14-s + (0.983 + 0.178i)15-s + (−0.963 + 0.266i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.753 + 0.657i)2-s + (0.473 − 0.880i)3-s + (0.134 + 0.990i)4-s + (0.309 + 0.951i)5-s + (0.936 − 0.351i)6-s + (−0.393 − 0.919i)7-s + (−0.550 + 0.834i)8-s + (−0.550 − 0.834i)9-s + (−0.393 + 0.919i)10-s + (−0.0448 + 0.998i)11-s + (0.936 + 0.351i)12-s + (−0.0448 − 0.998i)13-s + (0.309 − 0.951i)14-s + (0.983 + 0.178i)15-s + (−0.963 + 0.266i)16-s + (−0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.845 + 0.534i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ 0.845 + 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.422450850 + 0.4117124913i\)
\(L(\frac12)\) \(\approx\) \(1.422450850 + 0.4117124913i\)
\(L(1)\) \(\approx\) \(1.512205950 + 0.3332041402i\)
\(L(1)\) \(\approx\) \(1.512205950 + 0.3332041402i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.753 + 0.657i)T \)
3 \( 1 + (0.473 - 0.880i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.393 - 0.919i)T \)
11 \( 1 + (-0.0448 + 0.998i)T \)
13 \( 1 + (-0.0448 - 0.998i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.983 - 0.178i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (-0.691 - 0.722i)T \)
31 \( 1 + (-0.963 - 0.266i)T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (0.858 - 0.512i)T \)
47 \( 1 + (0.473 + 0.880i)T \)
53 \( 1 + (0.134 - 0.990i)T \)
59 \( 1 + (0.936 + 0.351i)T \)
61 \( 1 + (-0.393 + 0.919i)T \)
67 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (0.753 + 0.657i)T \)
79 \( 1 + (-0.550 + 0.834i)T \)
83 \( 1 + (0.936 + 0.351i)T \)
89 \( 1 + (0.134 - 0.990i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−31.52652056163862553455855979308, −31.052675042737716976617367671329, −29.153971903754616763052075546655, −28.566620241027623353423842809921, −27.58323183034576452473247896578, −26.19997628244798317320669197404, −24.78448782273356175954536043146, −24.0507328675491583896490533016, −22.21723423209933619630059602024, −21.703942360719768862615663966226, −20.7264113820339913329491100674, −19.754345924734323676363080977921, −18.64017648510016766696808234382, −16.51235061424571180576308684084, −15.74221136083672873889750051683, −14.39914298915943200812610975110, −13.38820318329593414389074234473, −12.1485201139373237627938318880, −10.875766934575846827635494087343, −9.40266918251698115807705262049, −8.762710444176358579480518660159, −6.00253787576065879325161432618, −4.94473771624652952970044834809, −3.62640442262391043360969168289, −2.13376972534334192415471982808, 2.478722138741004194894679294603, 3.73092196393313027007882215962, 5.77361629598589785179374610704, 7.18785375545918832911627151252, 7.495076765868605999423724060130, 9.55233169134642295449894361456, 11.32634155875295468883083073614, 12.8491110163384312790461984385, 13.62097351260479534035808622347, 14.59175982200043487406722495772, 15.667568998673848427972259769031, 17.490685688692484020352002696854, 18.01211110672806523271968185728, 19.763166879939363140793314123, 20.658017430275224396307362490807, 22.4686466652952705940667614334, 22.927784184407709181246463785801, 24.147785994473462622548287440059, 25.304359672116748344652381087582, 25.97107659933591637156975326704, 26.9007858788897712287060333895, 29.14988184719973245368962852461, 30.01012607283993952101330415046, 30.64324776817436957713279962887, 31.69170164193826853569101994512

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