Properties

Degree 1
Conductor 131
Sign $0.696 - 0.717i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.989 + 0.144i)2-s + (−0.527 + 0.849i)3-s + (0.958 − 0.285i)4-s + (0.926 − 0.377i)5-s + (0.399 − 0.916i)6-s + (0.215 − 0.976i)7-s + (−0.906 + 0.421i)8-s + (−0.443 − 0.896i)9-s + (−0.861 + 0.506i)10-s + (−0.262 − 0.964i)11-s + (−0.262 + 0.964i)12-s + (−0.861 − 0.506i)13-s + (−0.0724 + 0.997i)14-s + (−0.168 + 0.985i)15-s + (0.836 − 0.548i)16-s + (−0.998 − 0.0483i)17-s + ⋯
L(s,χ)  = 1  + (−0.989 + 0.144i)2-s + (−0.527 + 0.849i)3-s + (0.958 − 0.285i)4-s + (0.926 − 0.377i)5-s + (0.399 − 0.916i)6-s + (0.215 − 0.976i)7-s + (−0.906 + 0.421i)8-s + (−0.443 − 0.896i)9-s + (−0.861 + 0.506i)10-s + (−0.262 − 0.964i)11-s + (−0.262 + 0.964i)12-s + (−0.861 − 0.506i)13-s + (−0.0724 + 0.997i)14-s + (−0.168 + 0.985i)15-s + (0.836 − 0.548i)16-s + (−0.998 − 0.0483i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.696 - 0.717i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.696 - 0.717i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $0.696 - 0.717i$
motivic weight  =  \(0\)
character  :  $\chi_{131} (49, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 131,\ (0:\ ),\ 0.696 - 0.717i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5402251447 - 0.2283947786i$
$L(\frac12,\chi)$  $\approx$  $0.5402251447 - 0.2283947786i$
$L(\chi,1)$  $\approx$  0.6368148727 - 0.04534054064i
$L(1,\chi)$  $\approx$  0.6368148727 - 0.04534054064i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−28.83447713974385428747477482586, −28.131008466001271420997058240172, −26.86535068668258915829220292870, −25.63888208676351711878498564624, −24.99469278326303124142578004500, −24.27036969708858063289966056672, −22.70511347225949910867053255515, −21.70785955377068065900919995318, −20.68644202509834549988035068160, −19.19605152654440737922829415115, −18.63188128487529618947923603851, −17.50142092647623713779150802253, −17.26633779605950754745305096239, −15.62946384806228214551094968211, −14.44332755775088557235962597710, −12.85154434877996587934518417091, −12.05066399023356858742164801409, −10.91714245870969788934268418383, −9.79639106588601995896528776889, −8.67834521607245323224256992462, −7.29580093164795566749867251267, −6.451581592573162456851854538383, −5.260781121011951165106336590399, −2.41005703294395835538778676895, −1.87416264805960868350366646735, 0.765251657143263399359801847832, 2.75454569761601704085136809523, 4.71069652765433562153292531796, 5.86498045770373557696198052491, 7.05871111855037519920044372732, 8.66618799249096576346908426535, 9.54412315213128939930060231680, 10.62573840646067564249197814595, 11.15760487000485871340586227309, 12.90202071381430214745729707320, 14.32990496136506680329482517806, 15.53160049076498832301687483828, 16.63420512474591332054560251170, 17.24425830559162417796209231913, 17.92596652714591110016924629617, 19.536516197521958669744865681524, 20.53249108284062974299100034562, 21.286031290790345818879224916118, 22.355765471947993248775918645981, 23.886453446568264128859246669068, 24.57704719838456438916250006150, 26.004355256620925402403359056499, 26.63374774313124691728366072264, 27.4545826368620802628586601643, 28.49265450066073200226739705672

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