Properties

Degree 1
Conductor 137
Sign $-0.898 - 0.439i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.273 + 0.961i)2-s + (0.183 + 0.982i)3-s + (−0.850 + 0.526i)4-s + (−0.961 − 0.273i)5-s + (−0.895 + 0.445i)6-s + (−0.0922 + 0.995i)7-s + (−0.739 − 0.673i)8-s + (−0.932 + 0.361i)9-s i·10-s + (0.850 − 0.526i)11-s + (−0.673 − 0.739i)12-s + (−0.995 − 0.0922i)13-s + (−0.982 + 0.183i)14-s + (0.0922 − 0.995i)15-s + (0.445 − 0.895i)16-s + (−0.739 + 0.673i)17-s + ⋯
L(s,χ)  = 1  + (0.273 + 0.961i)2-s + (0.183 + 0.982i)3-s + (−0.850 + 0.526i)4-s + (−0.961 − 0.273i)5-s + (−0.895 + 0.445i)6-s + (−0.0922 + 0.995i)7-s + (−0.739 − 0.673i)8-s + (−0.932 + 0.361i)9-s i·10-s + (0.850 − 0.526i)11-s + (−0.673 − 0.739i)12-s + (−0.995 − 0.0922i)13-s + (−0.982 + 0.183i)14-s + (0.0922 − 0.995i)15-s + (0.445 − 0.895i)16-s + (−0.739 + 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.898 - 0.439i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.898 - 0.439i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(137\)
\( \varepsilon \)  =  $-0.898 - 0.439i$
motivic weight  =  \(0\)
character  :  $\chi_{137} (8, \cdot )$
Sato-Tate  :  $\mu(68)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 137,\ (0:\ ),\ -0.898 - 0.439i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.1692313395 + 0.7307380922i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.1692313395 + 0.7307380922i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4529300230 + 0.7277431858i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4529300230 + 0.7277431858i\)

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

−28.09127479025539181400271243892, −26.99160024923613675495133439393, −26.26176308645225541669990328390, −24.551245847625496343659499905883, −23.81128432678199264955980435196, −22.84763808100783846771925602209, −22.2386715287000280596692211990, −20.39537409909623008211593371589, −19.79775378477091606631395933908, −19.29792009125864188763901938515, −17.99791818646696435465196524936, −17.1670956215324843460815495686, −15.315264026136443352088911607140, −14.136685068752079489969818799188, −13.47549054589497843791914406271, −12.00178081610014103591510109615, −11.74322099958396545067058018974, −10.295633221844805359906765654317, −8.9866805061076397013290425131, −7.56994755931491795736793732668, −6.711148477014540012358441186487, −4.72044159954461745406916685307, −3.59340116234635353419375094659, −2.27030889183531427670450779535, −0.60648496489733105278161590854, 3.11852187625920370924806484986, 4.22174505657767835461159804401, 5.21082405872141706540075609928, 6.47286892059297183413390081309, 8.1600769532600335217937033756, 8.72588365684851876972907984890, 9.90169561307910404433228830252, 11.66228504510659380233097760961, 12.44876216902440230429350379051, 14.17864811663744010572722189666, 14.91871975644595922702101283788, 15.81756475765588136510926390423, 16.44303264699588233425540954072, 17.58033613967286944564325326731, 19.06635314059016688768258142825, 19.98792771475517407961835216658, 21.49540147284140012520104500294, 22.15321971535304235830565705107, 22.957681682378616851700255705136, 24.37743167425519835884094875375, 24.89209526222757416343650789101, 26.21964342583531895088812391049, 27.033801884139014462886594644443, 27.64433833385412002715463184756, 28.65612979914801940870058186650

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