Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $-0.554 + 0.832i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.690 − 0.723i)2-s + (0.0475 + 0.998i)3-s + (−0.0475 + 0.998i)4-s + (−0.755 + 0.654i)5-s + (0.690 − 0.723i)6-s + (−0.945 − 0.327i)7-s + (0.755 − 0.654i)8-s + (−0.995 + 0.0950i)9-s + (0.995 + 0.0950i)10-s + (0.945 − 0.327i)11-s − 12-s + (0.415 + 0.909i)14-s + (−0.690 − 0.723i)15-s + (−0.995 − 0.0950i)16-s + (−0.723 − 0.690i)17-s + (0.755 + 0.654i)18-s + ⋯
L(s,χ)  = 1  + (−0.690 − 0.723i)2-s + (0.0475 + 0.998i)3-s + (−0.0475 + 0.998i)4-s + (−0.755 + 0.654i)5-s + (0.690 − 0.723i)6-s + (−0.945 − 0.327i)7-s + (0.755 − 0.654i)8-s + (−0.995 + 0.0950i)9-s + (0.995 + 0.0950i)10-s + (0.945 − 0.327i)11-s − 12-s + (0.415 + 0.909i)14-s + (−0.690 − 0.723i)15-s + (−0.995 − 0.0950i)16-s + (−0.723 − 0.690i)17-s + (0.755 + 0.654i)18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.554 + 0.832i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.554 + 0.832i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $-0.554 + 0.832i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (210, \cdot )$
Sato-Tate  :  $\mu(132)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1157,\ (1:\ ),\ -0.554 + 0.832i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2261292774 + 0.4221399823i$
$L(\frac12,\chi)$  $\approx$  $0.2261292774 + 0.4221399823i$
$L(\chi,1)$  $\approx$  0.5550365624 + 0.06738422953i
$L(1,\chi)$  $\approx$  0.5550365624 + 0.06738422953i

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

−20.16761478597049266212671746828, −19.71676821719544491086084884045, −19.38489685113070825480207181604, −18.57752587254375796622869722988, −17.4548838392362065112204977395, −17.27272446912922829639368112977, −16.14291540588703512090756668445, −15.57938813204864370039114960166, −14.80609027234405407274295071880, −13.710435509565371783121380197955, −13.12132811182620277789531606523, −12.06505418589300780374384111799, −11.62740050692143960349630822050, −10.38800862329040040416077155308, −9.27183998978706059868993697866, −8.740402285039337003254810141728, −8.083452961524614339461077081425, −6.93782351397719749807528683893, −6.68988187793849001171341816355, −5.679427263765064086232070266975, −4.65040179120525024689145523330, −3.44800509894720337509936193456, −2.105736671420783065332338196037, −1.1115620279725450881113532000, −0.200536022770066767745179567646, 0.71429437935931090032439210806, 2.44162616233404353235208908124, 3.22774547477042831012688487766, 3.89602593307245625994052112595, 4.523190977700593284759365870945, 6.23523180161091537411945965449, 6.89450212566832321049178320286, 8.08093430217253696563283726019, 8.72817279594764299407275957501, 9.72863399355680532291448907428, 10.168701480038865247294260682579, 10.9986027626988983740200645815, 11.721932738780624666320240620180, 12.323671727477275036811734929386, 13.63948407484401680330693741716, 14.31393885372004834519478719992, 15.38570233527411408430614913131, 16.22734040518386213682472750434, 16.48965260018567430914765723891, 17.52145248449170047187410990685, 18.41487436519383898184292779408, 19.36797168777871874707598917418, 19.68971893616914616746478729496, 20.41572123887040692362986855028, 21.2557915832399950977652167257

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