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Properties

Label 1-131-131.48-r0-0-0
Degree $1$
Conductor $131$
Sign $0.999 - 0.0412i$
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  + (−0.861 − 0.506i)2-s + (0.836 + 0.548i)3-s + (0.485 + 0.873i)4-s + (0.779 − 0.626i)5-s + (−0.443 − 0.896i)6-s + (0.715 + 0.698i)7-s + (0.0241 − 0.999i)8-s + (0.399 + 0.916i)9-s + (−0.989 + 0.144i)10-s + (−0.0724 − 0.997i)11-s + (−0.0724 + 0.997i)12-s + (−0.989 − 0.144i)13-s + (−0.262 − 0.964i)14-s + (0.995 − 0.0965i)15-s + (−0.527 + 0.849i)16-s + (0.644 + 0.764i)17-s + ⋯
L(s)  = 1  + (−0.861 − 0.506i)2-s + (0.836 + 0.548i)3-s + (0.485 + 0.873i)4-s + (0.779 − 0.626i)5-s + (−0.443 − 0.896i)6-s + (0.715 + 0.698i)7-s + (0.0241 − 0.999i)8-s + (0.399 + 0.916i)9-s + (−0.989 + 0.144i)10-s + (−0.0724 − 0.997i)11-s + (−0.0724 + 0.997i)12-s + (−0.989 − 0.144i)13-s + (−0.262 − 0.964i)14-s + (0.995 − 0.0965i)15-s + (−0.527 + 0.849i)16-s + (0.644 + 0.764i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.999 - 0.0412i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.999 - 0.0412i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100065227 + 0.02271029593i\)
\(L(\frac12)\) \(\approx\) \(1.100065227 + 0.02271029593i\)
\(L(1)\) \(\approx\) \(1.054296210 - 0.04607278253i\)
\(L(1)\) \(\approx\) \(1.054296210 - 0.04607278253i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (-0.861 - 0.506i)T \)
3 \( 1 + (0.836 + 0.548i)T \)
5 \( 1 + (0.779 - 0.626i)T \)
7 \( 1 + (0.715 + 0.698i)T \)
11 \( 1 + (-0.0724 - 0.997i)T \)
13 \( 1 + (-0.989 - 0.144i)T \)
17 \( 1 + (0.644 + 0.764i)T \)
19 \( 1 + (-0.970 + 0.239i)T \)
23 \( 1 + (-0.943 + 0.331i)T \)
29 \( 1 + (0.399 - 0.916i)T \)
31 \( 1 + (0.958 - 0.285i)T \)
37 \( 1 + (0.981 - 0.192i)T \)
41 \( 1 + (-0.527 - 0.849i)T \)
43 \( 1 + (-0.998 + 0.0483i)T \)
47 \( 1 + (-0.748 + 0.663i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.926 - 0.377i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.998 - 0.0483i)T \)
71 \( 1 + (0.568 + 0.822i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (-0.906 + 0.421i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.168 - 0.985i)T \)
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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

−28.72328959804946542464460345882, −27.33082327098511792113644423751, −26.55692123412430891679674210205, −25.67358846785685404032618674797, −25.03556961749430225145610822780, −24.01647273109704291287057025811, −23.0649447605028682072611543671, −21.33397934108538790943689116698, −20.31276489715733791642436191695, −19.51163145539401072589470401483, −18.23662310891226061709466405288, −17.78840089446994044104344438894, −16.70325605634098620887426985382, −14.906087150711517391012846083470, −14.56994688363487800935237608671, −13.51391389949418826993906383399, −11.89265501860768235670195152503, −10.276293567321581217517405801600, −9.70515034398992633900665479131, −8.29149042228717883327795201597, −7.28432214606239477485863343042, −6.597109118314877515485236325187, −4.81393547919027831295544417223, −2.58770245140732897346990549612, −1.58670112229284701249091828869, 1.74610271234772891570167164335, 2.7361418978154641027863702032, 4.358142765537453231649652515531, 5.88957113398492685584198703851, 8.02208114708655337063186141086, 8.4941136676619055003468702806, 9.630738636170545124455322074087, 10.41907224364505362087373903161, 11.84732850677805228605953462774, 13.006000414911157925745520270, 14.233546830350241209139359229972, 15.42825604090624755889040842265, 16.630715912436359334589684468066, 17.42513825357325091454835809874, 18.738088909494061603984465065013, 19.560259440612155063058001975143, 20.67861809555183837244041182567, 21.52506855063969047320334319151, 21.78359999135748411498356753196, 24.26101023639953223074432604234, 24.95094801485836422475866615650, 25.78801711014573141982183394288, 26.82396534057159976583046407957, 27.68015665366469827102120879449, 28.42125816902780927846392175475

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