Properties

Degree 1
Conductor 101
Sign $-0.450 - 0.893i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.637 − 0.770i)2-s + (−0.535 + 0.844i)3-s + (−0.187 − 0.982i)4-s + (−0.637 − 0.770i)5-s + (0.309 + 0.951i)6-s + (−0.0627 − 0.998i)7-s + (−0.876 − 0.481i)8-s + (−0.425 − 0.904i)9-s − 10-s + (0.425 + 0.904i)11-s + (0.929 + 0.368i)12-s + (0.0627 − 0.998i)13-s + (−0.809 − 0.587i)14-s + (0.992 − 0.125i)15-s + (−0.929 + 0.368i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s,χ)  = 1  + (0.637 − 0.770i)2-s + (−0.535 + 0.844i)3-s + (−0.187 − 0.982i)4-s + (−0.637 − 0.770i)5-s + (0.309 + 0.951i)6-s + (−0.0627 − 0.998i)7-s + (−0.876 − 0.481i)8-s + (−0.425 − 0.904i)9-s − 10-s + (0.425 + 0.904i)11-s + (0.929 + 0.368i)12-s + (0.0627 − 0.998i)13-s + (−0.809 − 0.587i)14-s + (0.992 − 0.125i)15-s + (−0.929 + 0.368i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.450 - 0.893i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.450 - 0.893i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $-0.450 - 0.893i$
motivic weight  =  \(0\)
character  :  $\chi_{101} (23, \cdot )$
Sato-Tate  :  $\mu(50)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 101,\ (0:\ ),\ -0.450 - 0.893i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4859656993 - 0.7890573946i$
$L(\frac12,\chi)$  $\approx$  $0.4859656993 - 0.7890573946i$
$L(\chi,1)$  $\approx$  0.8439379608 - 0.5481786726i
$L(1,\chi)$  $\approx$  0.8439379608 - 0.5481786726i

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

−30.40859239306360295947787417093, −29.54340003335419864171631282432, −28.18168158613521821834932684707, −26.980956516648080627479599785614, −25.82361641218969474591174173440, −24.859198297493392571752668209988, −23.90597396807494701692123606649, −23.17172749856065062580346840570, −22.12316020370031596646757522408, −21.40330712337804676057474027910, −19.10238787609431923898285754596, −18.80261305043197592817633397732, −17.338172720023823818744382625, −16.39017837316304474719625655827, −15.14522010474179318329979266886, −14.231339610580352900070829661325, −12.96311459618298593676792339703, −11.89535461022352708047507147272, −11.167131218405078257802489735264, −8.76554785407493496957053369033, −7.729012494898740603376498308126, −6.45471291077171358945651719541, −5.8529609876562697234920275601, −4.05373489062476624057204375733, −2.49175638717995648124522578026, 0.86009151957033825889603939242, 3.33462325260433505507569997818, 4.44572723454846080743152135357, 5.161970456454513726245699249104, 6.937790131641340705945386389114, 8.93486950321751691081723018130, 10.116567859978001592565376960899, 11.013183466781095330957451589563, 12.14927201778239808067560896327, 13.04860372591849233744043221545, 14.59881612574662927397841838341, 15.536063601157193184540326424393, 16.69434929501268302781776790447, 17.800185955620528054702844595776, 19.5874278614521875062419384382, 20.37572450911845882047877188236, 20.957506626153343979613397881914, 22.4458292535321175451563297115, 23.092838418701600317604121487916, 23.82455416347764550755235043876, 25.34539562934147156586059348029, 27.22217406615699655511420596389, 27.465004509202513068776508446987, 28.54966270809360743469800831788, 29.502606901497445156599967201267

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