Properties

Degree 1
Conductor 71
Sign $0.564 - 0.825i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $0.564 - 0.825i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (15, \cdot )$
Sato-Tate  :  $\mu(35)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (0:\ ),\ 0.564 - 0.825i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7435696411 - 0.3923502619i$
$L(\frac12,\chi)$  $\approx$  $0.7435696411 - 0.3923502619i$
$L(\chi,1)$  $\approx$  0.8653195336 - 0.3033580507i
$L(1,\chi)$  $\approx$  0.8653195336 - 0.3033580507i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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