Properties

Degree 1
Conductor 101
Sign $-0.358 + 0.933i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.368 − 0.929i)2-s + (0.770 − 0.637i)3-s + (−0.728 − 0.684i)4-s + (−0.929 + 0.368i)5-s + (−0.309 − 0.951i)6-s + (−0.248 + 0.968i)7-s + (−0.904 + 0.425i)8-s + (0.187 − 0.982i)9-s + i·10-s + (−0.982 − 0.187i)11-s + (−0.998 − 0.0627i)12-s + (−0.968 + 0.248i)13-s + (0.809 + 0.587i)14-s + (−0.481 + 0.876i)15-s + (0.0627 + 0.998i)16-s + (−0.309 + 0.951i)17-s + ⋯
L(s,χ)  = 1  + (0.368 − 0.929i)2-s + (0.770 − 0.637i)3-s + (−0.728 − 0.684i)4-s + (−0.929 + 0.368i)5-s + (−0.309 − 0.951i)6-s + (−0.248 + 0.968i)7-s + (−0.904 + 0.425i)8-s + (0.187 − 0.982i)9-s + i·10-s + (−0.982 − 0.187i)11-s + (−0.998 − 0.0627i)12-s + (−0.968 + 0.248i)13-s + (0.809 + 0.587i)14-s + (−0.481 + 0.876i)15-s + (0.0627 + 0.998i)16-s + (−0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.358 + 0.933i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.358 + 0.933i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $-0.358 + 0.933i$
motivic weight  =  \(0\)
character  :  $\chi_{101} (67, \cdot )$
Sato-Tate  :  $\mu(100)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 101,\ (1:\ ),\ -0.358 + 0.933i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1713212057 - 0.2493077621i$
$L(\frac12,\chi)$  $\approx$  $-0.1713212057 - 0.2493077621i$
$L(\chi,1)$  $\approx$  0.6884265386 - 0.5314435987i
$L(1,\chi)$  $\approx$  0.6884265386 - 0.5314435987i

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

−30.88791436969998892352216486178, −29.538261177088505289699180697442, −27.74154533149091822205088413563, −26.92434054780919676930177130673, −26.35667798473457890573832510910, −25.204177025887435455630062254558, −24.15473956651765188919604124953, −23.18520244002722742351888822871, −22.266645351182361382457328633084, −20.84680707310130200955173161381, −20.08504135723608284045676261608, −18.8323979963552546944625335505, −17.22096062079654763629908636256, −16.11836496175399143801406687186, −15.58240324715592285464987332760, −14.39109023617925610210148699982, −13.440337210661004006484142583731, −12.258240375911065938997386186055, −10.42078702687472173749962313860, −9.20869278111105258718356749854, −7.79655306993418932759126831518, −7.39752530309884137694396487631, −5.17392847172358930076929142820, −4.23082970292227586617051500825, −3.126420716838632109860606604575, 0.10128410366150054797997830425, 2.28142249781666237640252886146, 3.06184741513054872555942836148, 4.5801618190040209236003515486, 6.34581227174576367139648179670, 7.95843332588837469809850637245, 8.955321428551191582760643894700, 10.35372317522832672954193715179, 11.81025341014693853972136207620, 12.4936888823575648345253675426, 13.58764906039322456269221545834, 14.91649081032910562142325618053, 15.48380118776246805465260852072, 17.8017242582582227429659225465, 18.927533392838164014899808932909, 19.27313735284500671351614627821, 20.358866664597117226231449780969, 21.54519367486899903292157830724, 22.53553197577379305915206475798, 23.836486253289024710047760593437, 24.379287702302668977299629617707, 26.10236073112547681028846614736, 26.79434676427674305802013275240, 28.219155669418339780379762177888, 28.986735375909148865987233285116

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