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Properties

Label 1-71-71.39-r1-0-0
Degree $1$
Conductor $71$
Sign $0.956 + 0.290i$
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  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)3-s + (0.623 + 0.781i)4-s + 5-s + (−0.222 − 0.974i)6-s + (0.900 − 0.433i)7-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.900 − 0.433i)10-s + (0.222 − 0.974i)11-s + (−0.222 + 0.974i)12-s + (0.222 + 0.974i)13-s − 14-s + (0.623 + 0.781i)15-s + (−0.222 + 0.974i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)3-s + (0.623 + 0.781i)4-s + 5-s + (−0.222 − 0.974i)6-s + (0.900 − 0.433i)7-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.900 − 0.433i)10-s + (0.222 − 0.974i)11-s + (−0.222 + 0.974i)12-s + (0.222 + 0.974i)13-s − 14-s + (0.623 + 0.781i)15-s + (−0.222 + 0.974i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.956 + 0.290i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ 0.956 + 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.694078596 + 0.2519349894i\)
\(L(\frac12)\) \(\approx\) \(1.694078596 + 0.2519349894i\)
\(L(1)\) \(\approx\) \(1.163060751 + 0.08256042143i\)
\(L(1)\) \(\approx\) \(1.163060751 + 0.08256042143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + T \)
7 \( 1 + (0.900 - 0.433i)T \)
11 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 - T \)
19 \( 1 + (0.623 - 0.781i)T \)
23 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (0.222 + 0.974i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (0.222 + 0.974i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 + (0.222 - 0.974i)T \)
61 \( 1 + (0.900 + 0.433i)T \)
67 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
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

−31.229665914514505627967511086554, −30.1472264763175816180122451558, −29.153569508320723716127508740721, −28.129022431056824599173392745173, −26.877650771741759114574954574781, −25.66739215516350011033823532739, −24.95619566793370865568443498729, −24.37532258216663507903866908803, −22.836160542549228843435434097334, −20.90594248701275354969251256795, −20.23765869136931479157421464901, −18.811948403040455697745379099814, −17.76605002989476664634826391932, −17.44683033865126729173533155526, −15.36141461964218104316297591424, −14.54057747198999502453545293172, −13.28517746523437557668425762416, −11.7351823703701299757841533991, −10.11130930084874528882268242164, −8.97255436957982651893870465268, −7.9274991195409092783715210300, −6.68288118566716111055837855952, −5.37553967569395301005579981738, −2.410860809064435512779234844629, −1.37116486061886103629375174419, 1.54887918675735184964209414641, 3.016427615757512382050398047408, 4.736155907737009358095322289621, 6.81475873575573068352023493431, 8.55870892912025571451151856393, 9.17501071167245113660175680436, 10.56469906892226900922882604661, 11.305472590856125511268569512743, 13.417214416135106545736081554599, 14.35138083778291092525072305929, 15.987472341278210436072809506022, 16.98252378684794731504494460750, 18.01955437394872994340500806297, 19.36164790269020775354167799590, 20.47626934090619324682207147398, 21.35590838584902938279269625647, 21.94639951703136309633164334586, 24.30101054291213079311284032270, 25.19722165709409668995853281840, 26.564165916127979011961992790900, 26.76399761835551453970817095427, 28.20046899730772833778834580137, 29.11849559766552703336671563810, 30.3483256646740340061622988004, 31.223272376840459264727208740652

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