Properties

Degree 1
Conductor 101
Sign $-0.144 - 0.989i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.684 + 0.728i)2-s + (−0.982 − 0.187i)3-s + (−0.0627 − 0.998i)4-s + (0.728 − 0.684i)5-s + (0.809 − 0.587i)6-s + (−0.481 + 0.876i)7-s + (0.770 + 0.637i)8-s + (0.929 + 0.368i)9-s + i·10-s + (0.368 − 0.929i)11-s + (−0.125 + 0.992i)12-s + (−0.876 + 0.481i)13-s + (−0.309 − 0.951i)14-s + (−0.844 + 0.535i)15-s + (−0.992 + 0.125i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s,χ)  = 1  + (−0.684 + 0.728i)2-s + (−0.982 − 0.187i)3-s + (−0.0627 − 0.998i)4-s + (0.728 − 0.684i)5-s + (0.809 − 0.587i)6-s + (−0.481 + 0.876i)7-s + (0.770 + 0.637i)8-s + (0.929 + 0.368i)9-s + i·10-s + (0.368 − 0.929i)11-s + (−0.125 + 0.992i)12-s + (−0.876 + 0.481i)13-s + (−0.309 − 0.951i)14-s + (−0.844 + 0.535i)15-s + (−0.992 + 0.125i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.144 - 0.989i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.144 - 0.989i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $-0.144 - 0.989i$
motivic weight  =  \(0\)
character  :  $\chi_{101} (55, \cdot )$
Sato-Tate  :  $\mu(100)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 101,\ (1:\ ),\ -0.144 - 0.989i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2933216444 - 0.3392613582i$
$L(\frac12,\chi)$  $\approx$  $0.2933216444 - 0.3392613582i$
$L(\chi,1)$  $\approx$  0.5367229983 + 0.009332826616i
$L(1,\chi)$  $\approx$  0.5367229983 + 0.009332826616i

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

−29.6903983219890998389384482096, −29.11098861820541132495052054639, −27.83700676157697374983125184362, −27.11291815808115722385461611335, −26.02035577898280299704231653999, −25.10949152539969270539004855018, −23.21223174547285543134734147639, −22.50812151507627575348758278543, −21.60023222754076450254919558283, −20.514696599308097054161436573489, −19.28079617023294481482492874035, −18.157397173101959396796761808144, −17.22903911558155359378166526333, −16.737806761230565190765161073214, −15.037469405954892014983302356874, −13.373449308661500058834072436376, −12.356120681414343133636212022450, −11.158451410459155688121006881499, −9.99739663719464033228947373860, −9.774180438248192097632519138524, −7.44410094769177758001522386856, −6.60448870140409579811681045775, −4.782386090504252647889569743248, −3.23155069056728306519921102490, −1.45839308289844756025382746341, 0.289441071069884612418251757350, 1.883495230396561137965902503268, 4.83622536075765917303664658403, 5.8901102931906749329725454384, 6.57959431337333662515240855283, 8.34123682243730080900426259639, 9.41973283995005777803007476515, 10.46242433411946351324067884931, 11.941161972070491717641428782948, 13.05242255100915102468805969315, 14.49626800907473027071825053504, 15.89532525819481526207725080635, 16.83293156934066879330920405636, 17.31408256590446320897136266718, 18.71923000193972350133664991452, 19.294400472513066220153748823361, 21.23943387955103487219603585179, 22.11826949595522886293865889727, 23.38659423505680430677915113156, 24.512975782584697497037899219165, 24.9042938073343231688647304133, 26.23652097225548953820439833515, 27.48801061758446232341122433795, 28.362019684337241182275572035066, 28.96623747278663312138768369039

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