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Properties

Label 1-131-131.43-r0-0-0
Degree $1$
Conductor $131$
Sign $-0.180 - 0.983i$
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  + (0.958 − 0.285i)2-s + (−0.443 − 0.896i)3-s + (0.836 − 0.548i)4-s + (0.715 − 0.698i)5-s + (−0.681 − 0.732i)6-s + (−0.906 − 0.421i)7-s + (0.644 − 0.764i)8-s + (−0.607 + 0.794i)9-s + (0.485 − 0.873i)10-s + (−0.861 + 0.506i)11-s + (−0.861 − 0.506i)12-s + (0.485 + 0.873i)13-s + (−0.989 − 0.144i)14-s + (−0.943 − 0.331i)15-s + (0.399 − 0.916i)16-s + (0.995 + 0.0965i)17-s + ⋯
L(s)  = 1  + (0.958 − 0.285i)2-s + (−0.443 − 0.896i)3-s + (0.836 − 0.548i)4-s + (0.715 − 0.698i)5-s + (−0.681 − 0.732i)6-s + (−0.906 − 0.421i)7-s + (0.644 − 0.764i)8-s + (−0.607 + 0.794i)9-s + (0.485 − 0.873i)10-s + (−0.861 + 0.506i)11-s + (−0.861 − 0.506i)12-s + (0.485 + 0.873i)13-s + (−0.989 − 0.144i)14-s + (−0.943 − 0.331i)15-s + (0.399 − 0.916i)16-s + (0.995 + 0.0965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $-0.180 - 0.983i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ -0.180 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.046030830 - 1.255951932i\)
\(L(\frac12)\) \(\approx\) \(1.046030830 - 1.255951932i\)
\(L(1)\) \(\approx\) \(1.296203424 - 0.8502102962i\)
\(L(1)\) \(\approx\) \(1.296203424 - 0.8502102962i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (0.958 - 0.285i)T \)
3 \( 1 + (-0.443 - 0.896i)T \)
5 \( 1 + (0.715 - 0.698i)T \)
7 \( 1 + (-0.906 - 0.421i)T \)
11 \( 1 + (-0.861 + 0.506i)T \)
13 \( 1 + (0.485 + 0.873i)T \)
17 \( 1 + (0.995 + 0.0965i)T \)
19 \( 1 + (-0.748 + 0.663i)T \)
23 \( 1 + (0.926 - 0.377i)T \)
29 \( 1 + (-0.607 - 0.794i)T \)
31 \( 1 + (-0.527 - 0.849i)T \)
37 \( 1 + (0.779 + 0.626i)T \)
41 \( 1 + (0.399 + 0.916i)T \)
43 \( 1 + (-0.168 + 0.985i)T \)
47 \( 1 + (0.568 + 0.822i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.215 + 0.976i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.168 - 0.985i)T \)
71 \( 1 + (-0.970 + 0.239i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (-0.998 + 0.0483i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.981 - 0.192i)T \)
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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

−29.21993872047512368695752525902, −28.16339186057106185222767387460, −26.707854557616097920598245950829, −25.7325991766003405488080771631, −25.28841428635499236306718870080, −23.495833152403400878309436016288, −22.91356328213393209353782898114, −21.85301919222097053274915033432, −21.43738517180715832664394039009, −20.33422688936059002968675378440, −18.76052367735828523681639349904, −17.458361565005652377213907659352, −16.38040103109688818172447987506, −15.51919877097515742239913106063, −14.7210485965859328519953037572, −13.432914771218501806749415463, −12.51867283372132473821632137423, −10.98182144902169322588632309056, −10.388259792191617289526496085379, −8.92049356262171042474436899472, −7.0930456019951631031879803089, −5.81266795943448966064358153898, −5.38671402864911866162179285495, −3.502959589797792426645804751, −2.793928062074951632150877194426, 1.33460853599944136407992667848, 2.61408396982158944363446314354, 4.39104061568446600953396732161, 5.71535292249164807249273448974, 6.44559579292465289117468625166, 7.739880802783920419584343332866, 9.62450169777391027709943043513, 10.73439504095040279492107559400, 12.07746615060332419776293396338, 13.04817817851802640568100109750, 13.34260748788109153015635688747, 14.66818958095825221260384434012, 16.377994077096175736259721321474, 16.82694402432905601226400841835, 18.47994491113750296726024143475, 19.3232504517991569769920331481, 20.56452206404945705054941216201, 21.315168170344740359628906313160, 22.68203441578427376142794625479, 23.35324054336706038104093967831, 24.135533291646665720532222098753, 25.30019419191056001055925844623, 25.840424799272518406151535676039, 28.07164727382379647034380486519, 28.797143893541040314793621471420

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