Properties

Degree 1
Conductor $ 2^{3} \cdot 11 $
Sign $0.286 - 0.958i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.809 − 0.587i)3-s + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)17-s + (0.809 − 0.587i)19-s − 21-s + 23-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (0.309 − 0.951i)31-s + (−0.309 + 0.951i)35-s + ⋯
L(s,χ)  = 1  + (0.809 − 0.587i)3-s + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)17-s + (0.809 − 0.587i)19-s − 21-s + 23-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (0.309 − 0.951i)31-s + (−0.309 + 0.951i)35-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.286 - 0.958i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.286 - 0.958i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(88\)    =    \(2^{3} \cdot 11\)
\( \varepsilon \)  =  $0.286 - 0.958i$
motivic weight  =  \(0\)
character  :  $\chi_{88} (69, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 88,\ (0:\ ),\ 0.286 - 0.958i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8924441364 - 0.6647488628i$
$L(\frac12,\chi)$  $\approx$  $0.8924441364 - 0.6647488628i$
$L(\chi,1)$  $\approx$  1.070335140 - 0.4416397990i
$L(1,\chi)$  $\approx$  1.070335140 - 0.4416397990i

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

−30.96191964039900575574183467361, −29.913239163543869937581636170816, −28.69003486131820503588897714705, −27.21088758345364092026153221085, −26.77453164151395179562237479132, −25.42602144417262692485651552783, −24.9825820647005053867901084134, −22.96870876150210591038124817216, −22.34416961249357682094333326093, −21.226582547745601694180554337345, −19.95721440845452326697021484293, −19.10376116275445776700145196243, −18.109124210828626649525328012920, −16.27589943210057285571709482550, −15.427248049936141145778838601607, −14.546423466434691579883742230, −13.360606142722183885976574822740, −11.88544306057441810066005663096, −10.39037058496070173849432723213, −9.57894225416485948505504232160, −8.15400384848128938660763966813, −6.936424135767199980785170805793, −5.25115521320241938508874235625, −3.40682081031153267409199794373, −2.72019739626508544739245282292, 1.28552444624621651567010587123, 3.175516719878358544055146768572, 4.51133607824761180530172099578, 6.48038819219605510860352280455, 7.64091074716715577527103382974, 8.86010485465199390682801119306, 9.81122865774917106542866881623, 11.75202855783698179707577079018, 12.88485344474850482489505257668, 13.60268726234987235537203376566, 14.97267548479408369099770213601, 16.261586481695591093279243231473, 17.24625746022053531305578073163, 18.85703358573236135433717200181, 19.6636378653633855374064291715, 20.4303546605954520208120001332, 21.64331977010997992429667588101, 23.33867631647104698925977857358, 24.010945237096412921367904895978, 25.067394626625363077491365757790, 26.106232807039725285688469647187, 26.98040669708874576689223119347, 28.53712352388948727405972112080, 29.255210021947870510013318570085, 30.51429412513386121883114136964

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