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Properties

Label 1-131-131.46-r0-0-0
Degree $1$
Conductor $131$
Sign $-0.659 - 0.751i$
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  + (0.399 + 0.916i)2-s + (−0.262 + 0.964i)3-s + (−0.681 + 0.732i)4-s + (−0.998 + 0.0483i)5-s + (−0.989 + 0.144i)6-s + (−0.168 − 0.985i)7-s + (−0.943 − 0.331i)8-s + (−0.861 − 0.506i)9-s + (−0.443 − 0.896i)10-s + (−0.527 + 0.849i)11-s + (−0.527 − 0.849i)12-s + (−0.443 + 0.896i)13-s + (0.836 − 0.548i)14-s + (0.215 − 0.976i)15-s + (−0.0724 − 0.997i)16-s + (0.926 − 0.377i)17-s + ⋯
L(s)  = 1  + (0.399 + 0.916i)2-s + (−0.262 + 0.964i)3-s + (−0.681 + 0.732i)4-s + (−0.998 + 0.0483i)5-s + (−0.989 + 0.144i)6-s + (−0.168 − 0.985i)7-s + (−0.943 − 0.331i)8-s + (−0.861 − 0.506i)9-s + (−0.443 − 0.896i)10-s + (−0.527 + 0.849i)11-s + (−0.527 − 0.849i)12-s + (−0.443 + 0.896i)13-s + (0.836 − 0.548i)14-s + (0.215 − 0.976i)15-s + (−0.0724 − 0.997i)16-s + (0.926 − 0.377i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $-0.659 - 0.751i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ -0.659 - 0.751i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1911087873 + 0.4218619426i\)
\(L(\frac12)\) \(\approx\) \(-0.1911087873 + 0.4218619426i\)
\(L(1)\) \(\approx\) \(0.4078382040 + 0.5578463330i\)
\(L(1)\) \(\approx\) \(0.4078382040 + 0.5578463330i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (0.399 + 0.916i)T \)
3 \( 1 + (-0.262 + 0.964i)T \)
5 \( 1 + (-0.998 + 0.0483i)T \)
7 \( 1 + (-0.168 - 0.985i)T \)
11 \( 1 + (-0.527 + 0.849i)T \)
13 \( 1 + (-0.443 + 0.896i)T \)
17 \( 1 + (0.926 - 0.377i)T \)
19 \( 1 + (-0.970 + 0.239i)T \)
23 \( 1 + (0.0241 + 0.999i)T \)
29 \( 1 + (-0.861 + 0.506i)T \)
31 \( 1 + (-0.607 + 0.794i)T \)
37 \( 1 + (-0.906 - 0.421i)T \)
41 \( 1 + (-0.0724 + 0.997i)T \)
43 \( 1 + (0.779 - 0.626i)T \)
47 \( 1 + (-0.748 + 0.663i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.644 + 0.764i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.779 + 0.626i)T \)
71 \( 1 + (0.568 + 0.822i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (0.981 + 0.192i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.715 + 0.698i)T \)
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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

−28.06228383547605654974792447334, −27.66024341881836222310084857835, −26.11810163369581813990397570293, −24.603456431552534892809784386627, −23.96014720372134601920500716073, −22.94258008959648833900766902426, −22.221509218441993510103831772746, −20.94374623421950521958031046630, −19.74813276611178080496279800185, −18.881410510659979996932226751575, −18.51611484480225595435282580474, −16.98279952694201696833130003551, −15.436193395199365322139031165303, −14.474584367732399555697439358809, −12.94914879849773792724210277137, −12.48058200704989770104706866548, −11.50317011657353727196840944420, −10.56093140502896220043241606503, −8.75050246739613999876670528032, −7.89028559679786965147643696947, −6.11237465917502974812276889466, −5.151865106735172587123646422720, −3.37960729061224899857925013103, −2.29607282053554703301063068654, −0.371074410043449841542600538842, 3.43803587618320381811612706330, 4.26615992194650883223117918496, 5.20921478825873562133257749947, 6.88811371602666004107015934250, 7.72155586897769820709017300418, 9.14137976981641879011699826332, 10.31551580944538242086020384833, 11.65022200369502312858155381686, 12.74750676980246871147728304200, 14.292938443966165006676138624340, 14.98169490822434487213500716322, 16.09838093452755117415423274371, 16.645769757242028840160867538123, 17.694158623048113189171645462960, 19.239325028243286591546728247171, 20.50460244432037190706380396950, 21.420350143307034387187257029901, 22.69568764191454614169133666701, 23.29880162226664545250204526566, 23.90962011391693767458747481712, 25.6776778281985180690651375321, 26.2505248103503211515511393042, 27.25892390247370181211960417869, 27.79286405125627588646717544215, 29.334661680744982253325226742272

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