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Properties

Label 1-71-71.15-r0-0-0
Degree $1$
Conductor $71$
Sign $0.564 - 0.825i$
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  + (−0.691 − 0.722i)2-s + (0.936 + 0.351i)3-s + (−0.0448 + 0.998i)4-s + (−0.809 − 0.587i)5-s + (−0.393 − 0.919i)6-s + (0.134 − 0.990i)7-s + (0.753 − 0.657i)8-s + (0.753 + 0.657i)9-s + (0.134 + 0.990i)10-s + (0.858 − 0.512i)11-s + (−0.393 + 0.919i)12-s + (0.858 + 0.512i)13-s + (−0.809 + 0.587i)14-s + (−0.550 − 0.834i)15-s + (−0.995 − 0.0896i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.691 − 0.722i)2-s + (0.936 + 0.351i)3-s + (−0.0448 + 0.998i)4-s + (−0.809 − 0.587i)5-s + (−0.393 − 0.919i)6-s + (0.134 − 0.990i)7-s + (0.753 − 0.657i)8-s + (0.753 + 0.657i)9-s + (0.134 + 0.990i)10-s + (0.858 − 0.512i)11-s + (−0.393 + 0.919i)12-s + (0.858 + 0.512i)13-s + (−0.809 + 0.587i)14-s + (−0.550 − 0.834i)15-s + (−0.995 − 0.0896i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ 0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7435696411 - 0.3923502619i\)
\(L(\frac12)\) \(\approx\) \(0.7435696411 - 0.3923502619i\)
\(L(1)\) \(\approx\) \(0.8653195336 - 0.3033580507i\)
\(L(1)\) \(\approx\) \(0.8653195336 - 0.3033580507i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.691 - 0.722i)T \)
3 \( 1 + (0.936 + 0.351i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.134 - 0.990i)T \)
11 \( 1 + (0.858 - 0.512i)T \)
13 \( 1 + (0.858 + 0.512i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.550 + 0.834i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-0.963 + 0.266i)T \)
31 \( 1 + (-0.995 + 0.0896i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (0.983 + 0.178i)T \)
47 \( 1 + (0.936 - 0.351i)T \)
53 \( 1 + (-0.0448 - 0.998i)T \)
59 \( 1 + (-0.393 + 0.919i)T \)
61 \( 1 + (0.134 + 0.990i)T \)
67 \( 1 + (-0.0448 + 0.998i)T \)
73 \( 1 + (-0.691 - 0.722i)T \)
79 \( 1 + (0.753 - 0.657i)T \)
83 \( 1 + (-0.393 + 0.919i)T \)
89 \( 1 + (-0.0448 - 0.998i)T \)
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

−31.94366872753933050163757487036, −30.84213872052155063863297697241, −29.955578528457115796438307806726, −28.03790788201154277106147535154, −27.55955549789625085250962781285, −26.05838773338124043121675046294, −25.61895307361630269004875635463, −24.42800924735244848500527861202, −23.50577706229716927176448394644, −22.10843465052358290405047237268, −20.32543411904505744790512558081, −19.32841261460462131034570763870, −18.623288454841979647542897035354, −17.54870914959488309395610835792, −15.6554855782935165599026709111, −15.14232651225369582153939536962, −14.157109359850604095346936266, −12.43385956560453673616895573646, −10.88804475782143427151759480494, −9.27688208486270349922054018756, −8.36381752989068341255847740468, −7.31574182059903877522160890061, −6.05739804423780310667602983212, −3.85346175715369471527302115753, −1.959627665280522336140143222142, 1.438239156288128546541990668457, 3.56606479746210687609160681244, 4.20549046076342322814919260434, 7.26518466495952479939258362501, 8.33528511077324385418982479179, 9.23806302377847146106136002235, 10.60399113057798595067498395154, 11.77177473194322298367561706430, 13.213442610149010213595543302835, 14.32579250486108466273013086422, 16.18489241973703107258538167795, 16.6765156340261706222666668878, 18.552726721607091541617093759228, 19.51252589447461810755756706075, 20.382996740833004976736590917203, 20.98695017142319641021220713072, 22.50177226037095460768530215296, 24.01970187052678497840418921005, 25.3079311520078547603328632623, 26.42195648917250190006796784288, 27.28386178946145573913937153222, 27.88389001402128861679892692447, 29.513206988083508507960619214, 30.41743840390948270770719578074, 31.38422399394179844118942280767

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