Properties

Degree 1
Conductor $ 2^{2} \cdot 17 $
Sign $0.615 - 0.788i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + i·3-s i·5-s i·7-s − 9-s i·11-s + 13-s + 15-s + 19-s + 21-s i·23-s − 25-s i·27-s i·29-s + i·31-s + 33-s + ⋯
L(s,χ)  = 1  + i·3-s i·5-s i·7-s − 9-s i·11-s + 13-s + 15-s + 19-s + 21-s i·23-s − 25-s i·27-s i·29-s + i·31-s + 33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.615 - 0.788i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.615 - 0.788i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(68\)    =    \(2^{2} \cdot 17\)
\( \varepsilon \)  =  $0.615 - 0.788i$
motivic weight  =  \(0\)
character  :  $\chi_{68} (55, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 68,\ (1:\ ),\ 0.615 - 0.788i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.293981416 - 0.6313702366i$
$L(\frac12,\chi)$  $\approx$  $1.293981416 - 0.6313702366i$
$L(\chi,1)$  $\approx$  1.069483852 - 0.1316594787i
$L(1,\chi)$  $\approx$  1.069483852 - 0.1316594787i

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

−31.19559444140688640022615804647, −30.929219950162945460361133657336, −29.749482371121348150403982360519, −28.68493297618495005910273509840, −27.628837697732914860783203067055, −25.852252554606769834446460831213, −25.47151903785208751938072382777, −24.105963986878647741211207094179, −22.94632647562644474479041849255, −22.13586602738546729951655831281, −20.547801650479837540117087844464, −19.18593153970871648123940319231, −18.32204487067672604232720366769, −17.649255210671784825437838825749, −15.7078372595981289059737320025, −14.61753529637072223787744924223, −13.42740525115170186492771092930, −12.12781554313383510668055346607, −11.16176873284845709347399249957, −9.43533293449462584194823355126, −7.90816107576477216381762090213, −6.76103989494615066846100100432, −5.59935294247543643857742433126, −3.17905263225444310197545440854, −1.81608290297430549981426346890, 0.78233919288920358360480950598, 3.475735391769599439548187855580, 4.61269640474115657509676974199, 5.97612465424866988433426872112, 8.08986620244006576139547150554, 9.14930395663210594216591763837, 10.45946514679554343345051515272, 11.54650159754358228805767169968, 13.27788613785467286695970578567, 14.25472847274316575517761318527, 16.05758793563453061646486150187, 16.39406236648201133834654502430, 17.672597350460670690529759709557, 19.52701858731940035128283962589, 20.57580954342557871966573908114, 21.20299893993558589400375836132, 22.642670274743384327920355314854, 23.70617356122254363252477260965, 24.9073489960730760735370232207, 26.34627187893960837758241397193, 27.04404892110194535782280609531, 28.2336936494814445513709244482, 29.03704374903079457094954745912, 30.51375899159975768952073104272, 31.83757255275859575598680726257

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