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Properties

Label 1-131-131.101-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: $\chi_{131} (101, \cdot )$
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

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

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