Properties

Degree 1
Conductor $ 2^{2} \cdot 17 $
Sign $0.990 + 0.139i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.923 − 0.382i)3-s + (0.382 + 0.923i)5-s + (−0.382 + 0.923i)7-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)11-s i·13-s + (0.707 + 0.707i)15-s + (−0.707 − 0.707i)19-s + i·21-s + (0.923 + 0.382i)23-s + (−0.707 + 0.707i)25-s + (0.382 − 0.923i)27-s + (−0.382 − 0.923i)29-s + (−0.923 + 0.382i)31-s − 33-s + ⋯
L(s,χ)  = 1  + (0.923 − 0.382i)3-s + (0.382 + 0.923i)5-s + (−0.382 + 0.923i)7-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)11-s i·13-s + (0.707 + 0.707i)15-s + (−0.707 − 0.707i)19-s + i·21-s + (0.923 + 0.382i)23-s + (−0.707 + 0.707i)25-s + (0.382 − 0.923i)27-s + (−0.382 − 0.923i)29-s + (−0.923 + 0.382i)31-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.990 + 0.139i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.990 + 0.139i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(68\)    =    \(2^{2} \cdot 17\)
\( \varepsilon \)  =  $0.990 + 0.139i$
motivic weight  =  \(0\)
character  :  $\chi_{68} (11, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 68,\ (0:\ ),\ 0.990 + 0.139i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.186431766 + 0.08301146307i$
$L(\frac12,\chi)$  $\approx$  $1.186431766 + 0.08301146307i$
$L(\chi,1)$  $\approx$  1.266858080 + 0.04481115819i
$L(1,\chi)$  $\approx$  1.266858080 + 0.04481115819i

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

−31.8285636999183897498996958127, −31.16692603538614796850540816802, −29.682090066591258197550520293839, −28.69118805901459611811273902350, −27.41825685552648495305904977405, −26.271298745026114039455977405015, −25.534972299073245851009669397708, −24.295093733333689902480451238586, −23.23385937639716677090324936923, −21.5111277595416073733155780746, −20.72122329942169699607737924979, −19.887461641270633754168845580911, −18.63366000336412758404040675941, −16.88919882499021318299191546543, −16.16301527138405442173579819010, −14.68927304609614791484781787120, −13.4999909673281063675544923668, −12.71229317369753158893052350061, −10.60507814962296583569553924244, −9.55602340663997763471099439333, −8.44578506814807660121756211206, −7.07222121326173318983131689460, −5.006295085566735332340316001584, −3.80466797672796764801784803227, −1.93240083009867725939841402286, 2.369554972275987239337818312421, 3.22509841415370874762253247956, 5.62610539689233642788624556475, 7.00822033934335875904075077332, 8.32310002757336929974161013776, 9.58390750890875960194823559635, 10.89147544192250539013510834210, 12.68330982138315305169948149913, 13.55063079922134479825155003383, 14.96322092719022958293727649985, 15.56872149141232608401714135684, 17.68690319678594267803076467969, 18.643827471163087233822410374692, 19.38870712997565186755908824165, 20.904338430854355451347333911317, 21.82923560608968384525742716179, 23.10819784487299049686147326064, 24.52837094670946988389707886219, 25.54677068205026399008746057718, 26.14831334277807088889168347054, 27.40764176843798867535522072808, 28.97588124688417959307656389144, 29.88428850035032853174632863870, 30.91268124750830569480694434498, 31.812840524550624832792515064065

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