Properties

Degree 1
Conductor $ 11 \cdot 17 $
Sign $0.0457 + 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s − 12-s + (−0.809 + 0.587i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s + ⋯
L(s,χ)  = 1  + (−0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s − 12-s + (−0.809 + 0.587i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0457 + 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0457 + 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(187\)    =    \(11 \cdot 17\)
\( \varepsilon \)  =  $0.0457 + 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{187} (16, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 187,\ (0:\ ),\ 0.0457 + 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4491125050 + 0.4290175490i$
$L(\frac12,\chi)$  $\approx$  $0.4491125050 + 0.4290175490i$
$L(\chi,1)$  $\approx$  0.6325809975 + 0.1987415146i
$L(1,\chi)$  $\approx$  0.6325809975 + 0.1987415146i

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

−26.88344744134137247341506246665, −26.34694137788202367779598506251, −25.39775912690465607297417773031, −24.24838466874153363806727301955, −22.74386874341848855964868597046, −21.91185186824383924099081977953, −21.06355255224887953754739695052, −20.21248265112554502913770632397, −19.60894588824483727245627424104, −17.812247777054225520323067125378, −17.33961617980360074463321945409, −16.51310673542761519552130859633, −15.63793948935621971532196066730, −14.03327445474222278242975671181, −12.940523194102971660897325644907, −11.82187754432679011958275761340, −10.600640518631641662970539960461, −9.926005615837965841158644174702, −9.2394121097147367954589544947, −7.94685697407706865275178410249, −6.50471146826718539149526431041, −5.02013703405967672600126337700, −3.8719837892065451983335723681, −2.50091102733110429398830595294, −0.652993461999274289293731725537, 1.695236770454120627073157657, 2.582101109883165187851546286213, 5.342271608540210556794444145317, 6.119051427375923167652865909537, 6.946612093651667822146300072533, 8.056578894368905669213717660416, 9.26321964011571838903404875031, 10.19148538042744842532823019151, 11.47719823187278921504473910650, 12.47906168171702849561902201297, 13.938601468160339271492392099080, 14.566567417335837828339279318767, 15.90322425669586204750874918778, 17.034338447902551688682870194744, 17.75536711682748901638559666957, 18.78291611046730346575872210718, 18.99175172532589397781044223483, 20.42748770419177454661763632282, 21.97543973088385777798954584865, 22.6858976907953684739347469234, 24.0119797409530183339282937897, 24.66568244868988362441658623459, 25.525034191600289216370147715683, 26.10666956577327852359629051624, 27.407698028563366328216197904576

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