Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $-0.561 + 0.827i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.540 − 0.841i)2-s + (−0.349 + 0.936i)3-s + (−0.415 + 0.909i)4-s + (0.142 + 0.989i)5-s + (0.977 − 0.212i)6-s + (0.800 + 0.599i)7-s + (0.989 − 0.142i)8-s + (−0.755 − 0.654i)9-s + (0.755 − 0.654i)10-s + (0.989 + 0.142i)11-s + (−0.707 − 0.707i)12-s + (0.0713 − 0.997i)14-s + (−0.977 − 0.212i)15-s + (−0.654 − 0.755i)16-s + (−0.540 + 0.841i)17-s + (−0.142 + 0.989i)18-s + ⋯
L(s,χ)  = 1  + (−0.540 − 0.841i)2-s + (−0.349 + 0.936i)3-s + (−0.415 + 0.909i)4-s + (0.142 + 0.989i)5-s + (0.977 − 0.212i)6-s + (0.800 + 0.599i)7-s + (0.989 − 0.142i)8-s + (−0.755 − 0.654i)9-s + (0.755 − 0.654i)10-s + (0.989 + 0.142i)11-s + (−0.707 − 0.707i)12-s + (0.0713 − 0.997i)14-s + (−0.977 − 0.212i)15-s + (−0.654 − 0.755i)16-s + (−0.540 + 0.841i)17-s + (−0.142 + 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.561 + 0.827i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.561 + 0.827i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $-0.561 + 0.827i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (252, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 1157,\ (0:\ ),\ -0.561 + 0.827i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.4428737293 + 0.8353364059i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.4428737293 + 0.8353364059i\)
\(L(\chi,1)\)  \(\approx\)  \(0.7325077893 + 0.2761743457i\)
\(L(1,\chi)\)  \(\approx\)  \(0.7325077893 + 0.2761743457i\)

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

−20.73082024744267034128435014772, −20.06280210219055276679345465131, −19.30119198541315106455846507527, −18.621396197120234084654799380, −17.54237161493314195592828579142, −17.19696911581410325532434304250, −16.81111932257096519767869721372, −15.76274706293985989403907791746, −14.78590286203970241264591660887, −13.81274273811359335327896305422, −13.53638911101308471988905351436, −12.4497011412398014322624837092, −11.505939414124581090319726497919, −10.83944825333679616968235203452, −9.646668917667556483348434820744, −8.70079335801920069253875624737, −8.29128341210864629623120800081, −7.2715434964982818418172748426, −6.67575381592904038328493709201, −5.74300296870213756700459889768, −4.88409002326277271785565623314, −4.21158342665485912011364598255, −2.17166224735779364038275365310, −1.28528938436504990087448069590, −0.54401608764108134138180471185, 1.437058472698739466409898123480, 2.42800368011946746924613461742, 3.37045762880129633928022584936, 4.18489276447663316124838861280, 5.024258655335552355832188827064, 6.25016127128535888418775683154, 7.05681590054742638279725554888, 8.49612517186244911806137578917, 8.83269703591234518873765374187, 9.8485601012548050733913739341, 10.677443600281360157401796970362, 11.04866571187953590637791227089, 11.83382726761066582673623543715, 12.5704380115234457513204188794, 13.869481446335842120112789520612, 14.75363578889630206061490074996, 15.126109386271829222620041193025, 16.32718707510841719747378718500, 17.22523633235924777789081080645, 17.67772254908127014388712257762, 18.36732713361377974076535457841, 19.40034646219627596527460927411, 19.868328059469036245229788018096, 21.0526120719578868741555823565, 21.58858105714016934598610023012

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