Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $0.861 + 0.507i$
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 + (0.800 + 0.599i)3-s + (0.142 − 0.989i)4-s + (−0.841 + 0.540i)5-s + (0.997 − 0.0713i)6-s + (0.977 + 0.212i)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 + (0.877 − 0.479i)14-s + (−0.997 − 0.0713i)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 + (0.800 + 0.599i)3-s + (0.142 − 0.989i)4-s + (−0.841 + 0.540i)5-s + (0.997 − 0.0713i)6-s + (0.977 + 0.212i)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 + (0.877 − 0.479i)14-s + (−0.997 − 0.0713i)15-s + (−0.959 − 0.281i)16-s + (0.755 + 0.654i)17-s + (0.841 + 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.861 + 0.507i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.861 + 0.507i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $0.861 + 0.507i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (281, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 1157,\ (0:\ ),\ 0.861 + 0.507i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.707080873 + 0.7372648165i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.707080873 + 0.7372648165i\)
\(L(\chi,1)\)  \(\approx\)  \(1.886798257 + 0.03037235344i\)
\(L(1,\chi)\)  \(\approx\)  \(1.886798257 + 0.03037235344i\)

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

−21.0471653847706110246237413522, −20.5019123029242877397601182867, −20.02479868025362261050811349301, −18.623083006776960077456443093500, −18.398798214407989616583923508660, −17.13808277620036893374550879339, −16.39144394527580285796293814210, −15.70872157980980237223579299685, −14.7948536085993073528951040281, −14.25717407192858167610773584921, −13.551431628115430823278419108894, −12.7745514142040862268844963492, −11.88246543841849847216427578014, −11.51726812034161986321496274442, −10.07063215830426207903034487922, −8.57045445155593338195457219666, −8.33211925414298147984156560113, −7.65363843787075984319613630764, −6.93235347060559310611279233459, −5.7045181653949273237811727451, −4.92288184115099716613708603905, −3.91989635681375340048268204115, −3.27566678887496625141910130566, −2.17048031203548269820514532641, −0.82815794745547389036744150483, 1.46433709022500198229913158571, 2.43197596416054231281377558172, 3.20937328150988870059384610364, 4.06190159059588686286310210698, 4.78222390690500132941767334264, 5.50704010480591059831466254922, 7.01278513555727524789631456336, 7.748546218998599777043706408451, 8.594923347407927385474173135723, 9.71923798291469462324569258166, 10.392060231693126526172644276943, 11.145178503585068125780896936356, 11.834648678049955418741413653915, 12.72671253749443573711466618941, 13.67773880780927995620413490627, 14.51063025740393226188609697713, 14.93872999790822160399755949031, 15.531078810218729548232570533086, 16.2754230324168235029582673805, 17.850575455322612827357146622914, 18.41746190951239628796420530765, 19.45932801235505163939317443090, 19.88859319263079891782564681492, 20.56315874779527258413976151446, 21.48252916075156047511078136731

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