Properties

Degree 1
Conductor 131
Sign $0.927 - 0.374i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.995 − 0.0965i)2-s + (0.779 − 0.626i)3-s + (0.981 − 0.192i)4-s + (−0.262 + 0.964i)5-s + (0.715 − 0.698i)6-s + (−0.989 − 0.144i)7-s + (0.958 − 0.285i)8-s + (0.215 − 0.976i)9-s + (−0.168 + 0.985i)10-s + (0.644 + 0.764i)11-s + (0.644 − 0.764i)12-s + (−0.168 − 0.985i)13-s + (−0.998 − 0.0483i)14-s + (0.399 + 0.916i)15-s + (0.926 − 0.377i)16-s + (−0.527 + 0.849i)17-s + ⋯
L(s,χ)  = 1  + (0.995 − 0.0965i)2-s + (0.779 − 0.626i)3-s + (0.981 − 0.192i)4-s + (−0.262 + 0.964i)5-s + (0.715 − 0.698i)6-s + (−0.989 − 0.144i)7-s + (0.958 − 0.285i)8-s + (0.215 − 0.976i)9-s + (−0.168 + 0.985i)10-s + (0.644 + 0.764i)11-s + (0.644 − 0.764i)12-s + (−0.168 − 0.985i)13-s + (−0.998 − 0.0483i)14-s + (0.399 + 0.916i)15-s + (0.926 − 0.377i)16-s + (−0.527 + 0.849i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.927 - 0.374i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.927 - 0.374i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $0.927 - 0.374i$
motivic weight  =  \(0\)
character  :  $\chi_{131} (33, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 131,\ (0:\ ),\ 0.927 - 0.374i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.108116933 - 0.4093523618i$
$L(\frac12,\chi)$  $\approx$  $2.108116933 - 0.4093523618i$
$L(\chi,1)$  $\approx$  1.953591376 - 0.2812966121i
$L(1,\chi)$  $\approx$  1.953591376 - 0.2812966121i

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

−28.8998031345091356102812356773, −27.80337655790294894958040604010, −26.55530220953162693851410318096, −25.54983264831017879876288800936, −24.7282778494350191362989932541, −23.85705851042178301140717027983, −22.56473250555930865226477854794, −21.63628733740657749792549591424, −20.91756369660499019342339220687, −19.63618661037599300421603349699, −19.40161427468903734979000041716, −16.86888068047340820389727334934, −16.15201920911588278856519085615, −15.483573209798811709493782595099, −14.10429245847001133485846616399, −13.40666644080727820204831481186, −12.2862951027957722407382155484, −11.13998362459460715421297086144, −9.522587446694000651537836791866, −8.67425173609981897895152721458, −7.17159594444528977799834452547, −5.76662714822539905861024962687, −4.388601947330880284772936237712, −3.64844996259312849227616216049, −2.1857579470467257737742876871, 2.05588506235637722472105973264, 3.170969526354813913641984215131, 4.07623579348195302329662111491, 6.26815690001629871930029590428, 6.81312080506186789702103230281, 8.00321348683093478208456975754, 9.80829084358589613533016194620, 10.87697100728873943389206396985, 12.49758147765979565128638811968, 12.86042670957559529667883501383, 14.30328203128000892621896818988, 14.86294296217882776305074420377, 15.82859407001456062436311302066, 17.48372211193943664429502511181, 18.88622962401384071863218044454, 19.70386784847463965122365837208, 20.320078535898191806529564274696, 21.86915068763558670365312243806, 22.66352194435683793488983655707, 23.46024699667213666577305507993, 24.6149649192939427733756128011, 25.70261083185032673039099903131, 26.02839937598035387093080274010, 27.675749297020640713912899878207, 29.20178277347223278612128251682

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