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Properties

Label 1-71-71.49-r0-0-0
Degree $1$
Conductor $71$
Sign $0.994 + 0.105i$
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  + (0.473 + 0.880i)2-s + (−0.0448 − 0.998i)3-s + (−0.550 + 0.834i)4-s + (0.309 − 0.951i)5-s + (0.858 − 0.512i)6-s + (0.983 + 0.178i)7-s + (−0.995 − 0.0896i)8-s + (−0.995 + 0.0896i)9-s + (0.983 − 0.178i)10-s + (0.753 − 0.657i)11-s + (0.858 + 0.512i)12-s + (0.753 + 0.657i)13-s + (0.309 + 0.951i)14-s + (−0.963 − 0.266i)15-s + (−0.393 − 0.919i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.473 + 0.880i)2-s + (−0.0448 − 0.998i)3-s + (−0.550 + 0.834i)4-s + (0.309 − 0.951i)5-s + (0.858 − 0.512i)6-s + (0.983 + 0.178i)7-s + (−0.995 − 0.0896i)8-s + (−0.995 + 0.0896i)9-s + (0.983 − 0.178i)10-s + (0.753 − 0.657i)11-s + (0.858 + 0.512i)12-s + (0.753 + 0.657i)13-s + (0.309 + 0.951i)14-s + (−0.963 − 0.266i)15-s + (−0.393 − 0.919i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.994 + 0.105i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ 0.994 + 0.105i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.184038049 + 0.06291793847i\)
\(L(\frac12)\) \(\approx\) \(1.184038049 + 0.06291793847i\)
\(L(1)\) \(\approx\) \(1.256906385 + 0.1102192132i\)
\(L(1)\) \(\approx\) \(1.256906385 + 0.1102192132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.473 + 0.880i)T \)
3 \( 1 + (-0.0448 - 0.998i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.983 + 0.178i)T \)
11 \( 1 + (0.753 - 0.657i)T \)
13 \( 1 + (0.753 + 0.657i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.963 + 0.266i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (0.936 - 0.351i)T \)
31 \( 1 + (-0.393 + 0.919i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (-0.691 + 0.722i)T \)
47 \( 1 + (-0.0448 + 0.998i)T \)
53 \( 1 + (-0.550 - 0.834i)T \)
59 \( 1 + (0.858 + 0.512i)T \)
61 \( 1 + (0.983 - 0.178i)T \)
67 \( 1 + (-0.550 + 0.834i)T \)
73 \( 1 + (0.473 + 0.880i)T \)
79 \( 1 + (-0.995 - 0.0896i)T \)
83 \( 1 + (0.858 + 0.512i)T \)
89 \( 1 + (-0.550 - 0.834i)T \)
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

−31.50440029043190051676984058922, −30.52579225593235007861585147347, −29.7838951895367859379323712404, −28.28079361026155251640507048989, −27.53330608839428553316109544181, −26.6241370907837637082972103421, −25.25242826560443312374953101853, −23.52852126435364822025731558031, −22.51057798678947092873283807862, −21.80806179385549089961591016156, −20.71350729827834263237992234911, −19.9127665137097805219738760294, −18.259702354975362746041452986577, −17.40965618646469640679494932259, −15.36765798528748599040816206801, −14.62942031102468047216879759200, −13.65296048630337976665836938077, −11.755530804827259309552291068849, −10.86418984826039314533317227190, −10.050337935162945312761094349016, −8.64774260339287589038661125712, −6.35096575576871352750619565844, −4.851381429390861092211466649558, −3.7355457939131297798431156120, −2.20053586064249887424669308267, 1.70989835138596392003310845740, 4.156065370619570312737105604376, 5.630304659920091628868588556, 6.61908231373028356011517609503, 8.34632354842535241143891526058, 8.71337936481880986785514199864, 11.468579882003709526929574578519, 12.499171053502140524542660844955, 13.64757508643507893809316696504, 14.38550242316569861504260336441, 16.05732353212506718422675586261, 17.2343734327727143947137491988, 17.862907069309452170891139160794, 19.35442059890172959253029986341, 20.90261226756482144959129345291, 21.87633901376806089552725085664, 23.51852566435053262353082382575, 24.10615792660242463776710055125, 24.86612313483257236886528789284, 25.80269078591597157699657274918, 27.346578884606361179467663460544, 28.47293076454629844114652637157, 29.87257647831876277484257275594, 30.79233366299381664102077419756, 31.69431283826966416385281468619

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