Properties

Label 1-131-131.33-r0-0-0
Degree $1$
Conductor $131$
Sign $0.927 - 0.374i$
Analytic cond. $0.608361$
Root an. cond. $0.608361$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.927 - 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.108116933 - 0.4093523618i\)
\(L(\frac12)\) \(\approx\) \(2.108116933 - 0.4093523618i\)
\(L(1)\) \(\approx\) \(1.953591376 - 0.2812966121i\)
\(L(1)\) \(\approx\) \(1.953591376 - 0.2812966121i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (0.995 - 0.0965i)T \)
3 \( 1 + (0.779 - 0.626i)T \)
5 \( 1 + (-0.262 + 0.964i)T \)
7 \( 1 + (-0.989 - 0.144i)T \)
11 \( 1 + (0.644 + 0.764i)T \)
13 \( 1 + (-0.168 - 0.985i)T \)
17 \( 1 + (-0.527 + 0.849i)T \)
19 \( 1 + (-0.970 + 0.239i)T \)
23 \( 1 + (-0.607 - 0.794i)T \)
29 \( 1 + (0.215 + 0.976i)T \)
31 \( 1 + (-0.943 - 0.331i)T \)
37 \( 1 + (-0.681 + 0.732i)T \)
41 \( 1 + (0.926 + 0.377i)T \)
43 \( 1 + (0.836 + 0.548i)T \)
47 \( 1 + (-0.748 + 0.663i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.0724 - 0.997i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.836 - 0.548i)T \)
71 \( 1 + (0.568 + 0.822i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (0.485 - 0.873i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.443 + 0.896i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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