Properties

Degree 1
Conductor 131
Sign $0.993 + 0.111i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.527 + 0.849i)2-s + (−0.681 + 0.732i)3-s + (−0.443 − 0.896i)4-s + (−0.906 − 0.421i)5-s + (−0.262 − 0.964i)6-s + (−0.998 + 0.0483i)7-s + (0.995 + 0.0965i)8-s + (−0.0724 − 0.997i)9-s + (0.836 − 0.548i)10-s + (0.958 − 0.285i)11-s + (0.958 + 0.285i)12-s + (0.836 + 0.548i)13-s + (0.485 − 0.873i)14-s + (0.926 − 0.377i)15-s + (−0.607 + 0.794i)16-s + (−0.943 − 0.331i)17-s + ⋯
L(s,χ)  = 1  + (−0.527 + 0.849i)2-s + (−0.681 + 0.732i)3-s + (−0.443 − 0.896i)4-s + (−0.906 − 0.421i)5-s + (−0.262 − 0.964i)6-s + (−0.998 + 0.0483i)7-s + (0.995 + 0.0965i)8-s + (−0.0724 − 0.997i)9-s + (0.836 − 0.548i)10-s + (0.958 − 0.285i)11-s + (0.958 + 0.285i)12-s + (0.836 + 0.548i)13-s + (0.485 − 0.873i)14-s + (0.926 − 0.377i)15-s + (−0.607 + 0.794i)16-s + (−0.943 − 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.993 + 0.111i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.993 + 0.111i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $0.993 + 0.111i$
motivic weight  =  \(0\)
character  :  $\chi_{131} (34, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 131,\ (0:\ ),\ 0.993 + 0.111i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4278841087 + 0.02391630655i$
$L(\frac12,\chi)$  $\approx$  $0.4278841087 + 0.02391630655i$
$L(\chi,1)$  $\approx$  0.4944454268 + 0.1574379404i
$L(1,\chi)$  $\approx$  0.4944454268 + 0.1574379404i

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

−28.70826681730539483131007674997, −27.84329762939666099158759706301, −26.990307824558960679673057176394, −25.805480578932903919063360916273, −24.817502340769411750772132261916, −23.14763034232129465536471795659, −22.776208576790666235082400565074, −21.85265764100963189725354049443, −20.101604659455434108437545285546, −19.493083198208884520639018771416, −18.66651317831343075814401285020, −17.71991927277105800176303343149, −16.64403831791857589366948425741, −15.61693120686234779996833275007, −13.717601246121332182214420377792, −12.73491419501761622424322782499, −11.819969154231400609100522495160, −11.05035967788346618548588210612, −9.87920207927859527134371909950, −8.409051349012439968625068817880, −7.29873010698415553524756944265, −6.24011956168346322191173582623, −4.18271180347444006385912916352, −3.02746630227844554171005197791, −1.22466464322981361717870256690, 0.614330732217260126112903508975, 3.74149107326241910226753561970, 4.748943047216150290941079128855, 6.20218088826420016045688765573, 6.98505131815753257009453808023, 8.82985651975983209847663830447, 9.27769127553340842651799901084, 10.77998246941948860595989698654, 11.71433930332935376078521250411, 13.20674776331864833124048650924, 14.70595852579618211802933627736, 15.87697686078151846089881922361, 16.19028816782926745004666524702, 17.12768824603528052782993248838, 18.40681716159475981891155955376, 19.49359138233137110843839338316, 20.39292245666346896481552308816, 22.16796745253046432401201427435, 22.74405309698082292601660478868, 23.75571581966769010315851596104, 24.61356268804387095086827315463, 26.03476045226714231074196618098, 26.70743449426032309980973606518, 27.65669705621152624464470699966, 28.402892888939414685631446884110

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