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  :  $\chi_{137} (120, \cdot )$
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

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

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