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Properties

Label 1-131-131.65-r0-0-0
Degree $1$
Conductor $131$
Sign $0.308 + 0.951i$
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  + (−0.998 + 0.0483i)2-s + (−0.943 + 0.331i)3-s + (0.995 − 0.0965i)4-s + (−0.607 − 0.794i)5-s + (0.926 − 0.377i)6-s + (−0.0724 + 0.997i)7-s + (−0.989 + 0.144i)8-s + (0.779 − 0.626i)9-s + (0.644 + 0.764i)10-s + (−0.906 − 0.421i)11-s + (−0.906 + 0.421i)12-s + (0.644 − 0.764i)13-s + (0.0241 − 0.999i)14-s + (0.836 + 0.548i)15-s + (0.981 − 0.192i)16-s + (0.485 + 0.873i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0483i)2-s + (−0.943 + 0.331i)3-s + (0.995 − 0.0965i)4-s + (−0.607 − 0.794i)5-s + (0.926 − 0.377i)6-s + (−0.0724 + 0.997i)7-s + (−0.989 + 0.144i)8-s + (0.779 − 0.626i)9-s + (0.644 + 0.764i)10-s + (−0.906 − 0.421i)11-s + (−0.906 + 0.421i)12-s + (0.644 − 0.764i)13-s + (0.0241 − 0.999i)14-s + (0.836 + 0.548i)15-s + (0.981 − 0.192i)16-s + (0.485 + 0.873i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.308 + 0.951i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.308 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3054879698 + 0.2220087278i\)
\(L(\frac12)\) \(\approx\) \(0.3054879698 + 0.2220087278i\)
\(L(1)\) \(\approx\) \(0.4512051114 + 0.09491781017i\)
\(L(1)\) \(\approx\) \(0.4512051114 + 0.09491781017i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (-0.998 + 0.0483i)T \)
3 \( 1 + (-0.943 + 0.331i)T \)
5 \( 1 + (-0.607 - 0.794i)T \)
7 \( 1 + (-0.0724 + 0.997i)T \)
11 \( 1 + (-0.906 - 0.421i)T \)
13 \( 1 + (0.644 - 0.764i)T \)
17 \( 1 + (0.485 + 0.873i)T \)
19 \( 1 + (0.120 + 0.992i)T \)
23 \( 1 + (-0.443 + 0.896i)T \)
29 \( 1 + (0.779 + 0.626i)T \)
31 \( 1 + (-0.168 + 0.985i)T \)
37 \( 1 + (0.399 + 0.916i)T \)
41 \( 1 + (0.981 + 0.192i)T \)
43 \( 1 + (0.958 + 0.285i)T \)
47 \( 1 + (-0.354 - 0.935i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.681 + 0.732i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.958 - 0.285i)T \)
71 \( 1 + (0.885 + 0.464i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.568 - 0.822i)T \)
83 \( 1 + (-0.861 + 0.506i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.527 - 0.849i)T \)
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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

−28.43690821533660071972036090465, −27.54489826398157564308533002208, −26.507656583145195959054661169, −25.95696898300917934004515580360, −24.36916349550586946799688897073, −23.49278511380693220069367123947, −22.77382182571935536333146497272, −21.31392781312456705394828596371, −20.17756350600123361942530438837, −19.01472654805749017403392975526, −18.317547347495039764064707622859, −17.454417591570640906885005038989, −16.2716028846343102738172207305, −15.67461193305878595678507952553, −14.00023268616010080643435797708, −12.50857678682785475180929237367, −11.28312423909089620857350526410, −10.78891053261754913778727593102, −9.71488398332463224841329244304, −7.81371640828040247942565818318, −7.18889506727964879050926069292, −6.212247269213404692882488175136, −4.31755586992075272179993404519, −2.50767143297420238601161541745, −0.60292090146784155001212943353, 1.275775953262322073397368025688, 3.391022382274095474113069927719, 5.33339264662598988082676788534, 6.040172129950863359668474284750, 7.8217406037406013148080821247, 8.63717645586506702054806942451, 9.95560681230289473949542553314, 10.95091609840722300915404453111, 12.03561769216656875722624268647, 12.73028370924643348896861666230, 15.14485123480115915051221439174, 15.91193407357581057222679227487, 16.48146087442546760716962753368, 17.76953086160277252722243689028, 18.500850580169040871005988089789, 19.602580911055488569510861271719, 20.91714936451238248743256694379, 21.517184639143463325477559309299, 23.14830323355120664348112682055, 23.93287649210937386497491889915, 24.94979668840585089480183895895, 26.02195726773380998828192712505, 27.45840898044820413740751184340, 27.690787732050291664292932649858, 28.70046973542731429532507121758

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