Properties

Degree 1
Conductor 71
Sign $0.992 + 0.118i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.936 + 0.351i)2-s + (0.858 − 0.512i)3-s + (0.753 + 0.657i)4-s + (−0.809 − 0.587i)5-s + (0.983 − 0.178i)6-s + (−0.550 + 0.834i)7-s + (0.473 + 0.880i)8-s + (0.473 − 0.880i)9-s + (−0.550 − 0.834i)10-s + (−0.691 − 0.722i)11-s + (0.983 + 0.178i)12-s + (−0.691 + 0.722i)13-s + (−0.809 + 0.587i)14-s + (−0.995 − 0.0896i)15-s + (0.134 + 0.990i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s,χ)  = 1  + (0.936 + 0.351i)2-s + (0.858 − 0.512i)3-s + (0.753 + 0.657i)4-s + (−0.809 − 0.587i)5-s + (0.983 − 0.178i)6-s + (−0.550 + 0.834i)7-s + (0.473 + 0.880i)8-s + (0.473 − 0.880i)9-s + (−0.550 − 0.834i)10-s + (−0.691 − 0.722i)11-s + (0.983 + 0.178i)12-s + (−0.691 + 0.722i)13-s + (−0.809 + 0.587i)14-s + (−0.995 − 0.0896i)15-s + (0.134 + 0.990i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.992 + 0.118i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.992 + 0.118i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $0.992 + 0.118i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (16, \cdot )$
Sato-Tate  :  $\mu(35)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (0:\ ),\ 0.992 + 0.118i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.648510356 + 0.09777596411i$
$L(\frac12,\chi)$  $\approx$  $1.648510356 + 0.09777596411i$
$L(\chi,1)$  $\approx$  1.712728870 + 0.09310965841i
$L(1,\chi)$  $\approx$  1.712728870 + 0.09310965841i

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

−31.77870460709192456011509619838, −30.5597688679744801881599880448, −30.09966638275489983746361902908, −28.53460777501220063085150345423, −27.24757076422856380831988464111, −26.197876336523399515201388190184, −25.18796238952029467449880739535, −23.658652961053090006991877059082, −22.87341997344554532439002554063, −21.78620346539179724013315766658, −20.51623460632305919033259952682, −19.76838300092031644081363353605, −18.922974423585427053390879194234, −16.66271115161748796155449951301, −15.21933751220095150002164503800, −14.924438348118775722994351594541, −13.44337246588533635596309737413, −12.46178060898309815164699603323, −10.61498597676676888480211524170, −10.15553659659031996704072763807, −7.94692462150839807651627691078, −6.82430597937355781691729221379, −4.71473923103069978711150585804, −3.66448595761880572870771595363, −2.55280809878436866489642947616, 2.45292903548172940519524044890, 3.67551042100373045775030852639, 5.26854294397934299022831162118, 6.89327839635338550943805437981, 8.03112371207408016476785220587, 9.14862172730208847528770384770, 11.549095461948454179654680899070, 12.54197240104520535081208973803, 13.37953179112894080189930180934, 14.74931917092987303822279145870, 15.65041798354645848275018666619, 16.66785738029829644401758849836, 18.69845288649671299220514563461, 19.579974176165163167892093523420, 20.77685539393446377798639333618, 21.72637349242175027684115394516, 23.2700451372622906611580627903, 24.08503657396288801250497906578, 24.95728245204885169724033517706, 25.91885575276254137467888529297, 27.118989214387573408195047000401, 28.88033173783698689341265656253, 29.744591466092853228480902901414, 31.202925332494410669037063188392, 31.72005812511298241684283993582

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