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Properties

Label 1-131-131.55-r0-0-0
Degree $1$
Conductor $131$
Sign $0.895 - 0.444i$
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  + (0.215 − 0.976i)2-s + (−0.998 + 0.0483i)3-s + (−0.906 − 0.421i)4-s + (0.836 + 0.548i)5-s + (−0.168 + 0.985i)6-s + (−0.443 + 0.896i)7-s + (−0.607 + 0.794i)8-s + (0.995 − 0.0965i)9-s + (0.715 − 0.698i)10-s + (0.926 − 0.377i)11-s + (0.926 + 0.377i)12-s + (0.715 + 0.698i)13-s + (0.779 + 0.626i)14-s + (−0.861 − 0.506i)15-s + (0.644 + 0.764i)16-s + (−0.0724 − 0.997i)17-s + ⋯
L(s)  = 1  + (0.215 − 0.976i)2-s + (−0.998 + 0.0483i)3-s + (−0.906 − 0.421i)4-s + (0.836 + 0.548i)5-s + (−0.168 + 0.985i)6-s + (−0.443 + 0.896i)7-s + (−0.607 + 0.794i)8-s + (0.995 − 0.0965i)9-s + (0.715 − 0.698i)10-s + (0.926 − 0.377i)11-s + (0.926 + 0.377i)12-s + (0.715 + 0.698i)13-s + (0.779 + 0.626i)14-s + (−0.861 − 0.506i)15-s + (0.644 + 0.764i)16-s + (−0.0724 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8910655898 - 0.2090816822i\)
\(L(\frac12)\) \(\approx\) \(0.8910655898 - 0.2090816822i\)
\(L(1)\) \(\approx\) \(0.8909629263 - 0.2468328884i\)
\(L(1)\) \(\approx\) \(0.8909629263 - 0.2468328884i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (0.215 - 0.976i)T \)
3 \( 1 + (-0.998 + 0.0483i)T \)
5 \( 1 + (0.836 + 0.548i)T \)
7 \( 1 + (-0.443 + 0.896i)T \)
11 \( 1 + (0.926 - 0.377i)T \)
13 \( 1 + (0.715 + 0.698i)T \)
17 \( 1 + (-0.0724 - 0.997i)T \)
19 \( 1 + (-0.970 + 0.239i)T \)
23 \( 1 + (0.958 + 0.285i)T \)
29 \( 1 + (0.995 + 0.0965i)T \)
31 \( 1 + (0.0241 - 0.999i)T \)
37 \( 1 + (0.485 + 0.873i)T \)
41 \( 1 + (0.644 - 0.764i)T \)
43 \( 1 + (-0.262 + 0.964i)T \)
47 \( 1 + (-0.748 + 0.663i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.527 + 0.849i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.262 - 0.964i)T \)
71 \( 1 + (0.568 + 0.822i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (-0.681 - 0.732i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.989 - 0.144i)T \)
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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

−28.58592724979860229887011918415, −27.77299233270897715877817794543, −26.72513826224180250046315835784, −25.48478336126444071110884702158, −24.83255808067241135469124393204, −23.57916839164694309174867862712, −23.05475803123061806879045627185, −21.98322234958566495358154408178, −21.10737129780919872033677090191, −19.53024251436540073297089348389, −17.995238373323279078805585157442, −17.22872534656323506280525394693, −16.7555674655252738926755326895, −15.653186515418640528462565566555, −14.29319288738431965053708008366, −13.0342958237393351466471223656, −12.6295186508080269160787109930, −10.743708899931699138886397680390, −9.71466161420852175066059403976, −8.42295433451580430068204320701, −6.7660676665368297814737620031, −6.24994130743513007510711765959, −4.97388043877035244297450448285, −3.92380474295306571339500484301, −1.08604786453105919006621221684, 1.47662750376290240863345708360, 2.93248996178314132380111089553, 4.494890360233207087299528662988, 5.86325900280645660474655534946, 6.51633035709085989905084402736, 8.99654983019734349425786804089, 9.751284354158942462810251138901, 11.01576254734892730164615487920, 11.68854621191481096698037398917, 12.83248177134787002854304840875, 13.839829920494202563087510025390, 15.075065744131695490634343240695, 16.56349912957569497157952386503, 17.66108750368095177508707749816, 18.595927589228031433155508561419, 19.18649604685194821192942070042, 21.02317090184543945714699876790, 21.60742167076779030208591930930, 22.444373308684282218468093367933, 23.10942486647660633981921158408, 24.486794802497894545355178273647, 25.65994490325575263284291977831, 27.06200645568653705441420227317, 27.86067308842923982079665548380, 28.90418141509999027605196501848

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