Properties

Degree 1
Conductor $ 3 \cdot 23 $
Sign $0.0174 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (0.654 − 0.755i)7-s + (0.142 − 0.989i)8-s + (0.654 + 0.755i)10-s + (0.841 − 0.540i)11-s + (−0.654 − 0.755i)13-s + (−0.959 + 0.281i)14-s + (−0.654 + 0.755i)16-s + (0.415 − 0.909i)17-s + (−0.415 − 0.909i)19-s + (−0.142 − 0.989i)20-s − 22-s + (0.841 + 0.540i)25-s + (0.142 + 0.989i)26-s + ⋯
L(s,χ)  = 1  + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (0.654 − 0.755i)7-s + (0.142 − 0.989i)8-s + (0.654 + 0.755i)10-s + (0.841 − 0.540i)11-s + (−0.654 − 0.755i)13-s + (−0.959 + 0.281i)14-s + (−0.654 + 0.755i)16-s + (0.415 − 0.909i)17-s + (−0.415 − 0.909i)19-s + (−0.142 − 0.989i)20-s − 22-s + (0.841 + 0.540i)25-s + (0.142 + 0.989i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0174 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0174 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(69\)    =    \(3 \cdot 23\)
\( \varepsilon \)  =  $0.0174 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{69} (5, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 69,\ (0:\ ),\ 0.0174 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4025361303 - 0.3955869120i$
$L(\frac12,\chi)$  $\approx$  $0.4025361303 - 0.3955869120i$
$L(\chi,1)$  $\approx$  0.5949408789 - 0.2848442032i
$L(1,\chi)$  $\approx$  0.5949408789 - 0.2848442032i

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

−32.172162349357803818836520340809, −31.10041439932825405211259233870, −29.92172134467860378015308594987, −28.39297722297770265929463489229, −27.65048919783941068303471336221, −26.82930028618820665035514230325, −25.61891761786768845703201945154, −24.50891740711789645953770663627, −23.65342752155071504312291473991, −22.37006472174596003399796238367, −20.780721871934710206591127482906, −19.405072946220673782092382510867, −18.80957058643508463842610847725, −17.458426386629060991123161798753, −16.41461823817766312923896813622, −14.95041345179794517896176548209, −14.64395095540596760351072159392, −12.14607366258038728997767902013, −11.270206793107503950819598859753, −9.741135844622269325723308305130, −8.44037817829674051986894639637, −7.45485707256871330040134563943, −6.06698271625630670537347745582, −4.32241025153315076046534592527, −1.956854261784113560006561567285, 0.96232529288932026425437074967, 3.17563253665816250225279113004, 4.58448608098081294817842887779, 7.09213010458559384774309357540, 8.02116693167184603254743865921, 9.250872197448729643811651275840, 10.81919639883970437045592058789, 11.61839443657721991344326816885, 12.84483996906254130795022635093, 14.55911894465751850506128753576, 16.08484076873491499065353741260, 17.01711152352820173562617645204, 18.135885167084961781797801264763, 19.61144919105697733286695821933, 20.03047585727433679253436896171, 21.31199272048150534504053371367, 22.658254742334272563299960294032, 24.047334194174373917445420764512, 25.06655135307265171234600307696, 26.63130559142641400312276172636, 27.303043872864401340587312627477, 27.98430271987152079814774581171, 29.56092660430121307856580182689, 30.208241459046460146719113953898, 31.3430845113279519303180029222

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