Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $-0.426 - 0.904i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.755 + 0.654i)2-s + (−0.599 + 0.800i)3-s + (0.142 − 0.989i)4-s + (−0.841 + 0.540i)5-s + (−0.0713 − 0.997i)6-s + (0.212 − 0.977i)7-s + (0.540 + 0.841i)8-s + (−0.281 − 0.959i)9-s + (0.281 − 0.959i)10-s + (0.540 − 0.841i)11-s + (0.707 + 0.707i)12-s + (0.479 + 0.877i)14-s + (0.0713 − 0.997i)15-s + (−0.959 − 0.281i)16-s + (−0.755 − 0.654i)17-s + (0.841 + 0.540i)18-s + ⋯
L(s,χ)  = 1  + (−0.755 + 0.654i)2-s + (−0.599 + 0.800i)3-s + (0.142 − 0.989i)4-s + (−0.841 + 0.540i)5-s + (−0.0713 − 0.997i)6-s + (0.212 − 0.977i)7-s + (0.540 + 0.841i)8-s + (−0.281 − 0.959i)9-s + (0.281 − 0.959i)10-s + (0.540 − 0.841i)11-s + (0.707 + 0.707i)12-s + (0.479 + 0.877i)14-s + (0.0713 − 0.997i)15-s + (−0.959 − 0.281i)16-s + (−0.755 − 0.654i)17-s + (0.841 + 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.426 - 0.904i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.426 - 0.904i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $-0.426 - 0.904i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (31, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 1157,\ (0:\ ),\ -0.426 - 0.904i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.08022210973 - 0.1264955262i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.08022210973 - 0.1264955262i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4466447681 + 0.1278147489i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4466447681 + 0.1278147489i\)

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

−21.67452650231793632164450083631, −20.38295489168958182565300879921, −19.76024654625734708728760722688, −19.37288112744125034125149116868, −18.282704974737959390298936336498, −17.897722876570836486009902644637, −17.19952297408701281203201076874, −16.17465241511247079517650192423, −15.72276221456041255590753798164, −14.54477365502068059410564808354, −13.26510020565535495740090538532, −12.46366048218178569899107796018, −12.14550836820540913983504415709, −11.41007940216877138556672668678, −10.76874340506104785797554020948, −9.451841684491126204241017065156, −8.81312650183304312600314395342, −7.968417521970029370581460776536, −7.313314116390513639934123751666, −6.41725930128178662919444863335, −5.186267398975245345633945441597, −4.34365629344167576449080923193, −3.15617682892281427168414505456, −1.96362975586959652263461818394, −1.36454640408492463398909446909, 0.103331047314007075199903531152, 1.10713810137383336645344050500, 2.90272367134361627715634130901, 4.07360253419586384223091924136, 4.58454457669698118722714279936, 5.901782951413747070652706613701, 6.481975024897937582350224770466, 7.45431648114578829558676514037, 8.11337263704722653138258372221, 9.171144832837855712360429103597, 9.96039884653239306093470191691, 10.71504719196968106125069743665, 11.36533052434243233229333519928, 11.84832242348449166636810310097, 13.60237729269386399904407009901, 14.30807867604061240927458765864, 15.010933001970623441318944734385, 15.89250909073070804315398699156, 16.35993344355245344339347453135, 16.98582682441112537034927390071, 17.89482116206864685008359822734, 18.49757349788329202792383113833, 19.48211480603605743119631342374, 20.18657640752264252379995674050, 20.75177388953303611396306139943

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