Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $0.596 - 0.802i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.971 + 0.235i)2-s + (−0.888 − 0.458i)3-s + (0.888 − 0.458i)4-s + (0.755 − 0.654i)5-s + (0.971 + 0.235i)6-s + (−0.189 − 0.981i)7-s + (−0.755 + 0.654i)8-s + (0.580 + 0.814i)9-s + (−0.580 + 0.814i)10-s + (0.189 − 0.981i)11-s − 12-s + (0.415 + 0.909i)14-s + (−0.971 + 0.235i)15-s + (0.580 − 0.814i)16-s + (−0.235 + 0.971i)17-s + (−0.755 − 0.654i)18-s + ⋯
L(s,χ)  = 1  + (−0.971 + 0.235i)2-s + (−0.888 − 0.458i)3-s + (0.888 − 0.458i)4-s + (0.755 − 0.654i)5-s + (0.971 + 0.235i)6-s + (−0.189 − 0.981i)7-s + (−0.755 + 0.654i)8-s + (0.580 + 0.814i)9-s + (−0.580 + 0.814i)10-s + (0.189 − 0.981i)11-s − 12-s + (0.415 + 0.909i)14-s + (−0.971 + 0.235i)15-s + (0.580 − 0.814i)16-s + (−0.235 + 0.971i)17-s + (−0.755 − 0.654i)18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.596 - 0.802i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.596 - 0.802i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $0.596 - 0.802i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (566, \cdot )$
Sato-Tate  :  $\mu(132)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1157,\ (1:\ ),\ 0.596 - 0.802i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9293637127 - 0.4674033631i$
$L(\frac12,\chi)$  $\approx$  $0.9293637127 - 0.4674033631i$
$L(\chi,1)$  $\approx$  0.6083499361 - 0.1675392802i
$L(1,\chi)$  $\approx$  0.6083499361 - 0.1675392802i

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

−21.05297310818561716808596846625, −20.70797061920726750442561216424, −19.39965133380680578968679891351, −18.66066037650615116517689487857, −18.06602036744831802582008630445, −17.39674483885526254931612707108, −16.93477021327602252599340542080, −15.7180287068631561794239508903, −15.37296883135955546326478200541, −14.51682761697063777161358911769, −12.98885755353790394945560350023, −12.41192214631813643987429782745, −11.41302442421430567958414408600, −10.97227194878588699211021324174, −10.00974921145365609039749542729, −9.39480721501788601504023032939, −8.915292467467647579475163846191, −7.307843489538891021410528554423, −6.80095034089146284838323483265, −5.93618026777418500818382752467, −5.140021603004477519174187508743, −3.83678868621195109455503537719, −2.57641841018501453001433896178, −2.00939314795011508676483287247, −0.58268460244372295648058207523, 0.60410632188806741018046713222, 1.24516977860443469175583894087, 2.107915482014514427690905836261, 3.6711919237416900216527716461, 4.99241533712948013111824590852, 5.83082056269535126141181424743, 6.483625233374589915942134242187, 7.2012364126179947916485671226, 8.26662524069928807699326726344, 8.917937593649906576498857089231, 10.01072871203191568575597290601, 10.70683543188581689156505393469, 11.12726350272607701348701106345, 12.39662011692151454414948408072, 12.991730513144550632425592165944, 13.88137597499096318788047056391, 14.81901141959773329707029077740, 16.13834144478688687605574002809, 16.651054683862484745398432218381, 17.03104400017985507982211138225, 17.712550230406208883808478506894, 18.51978200466955548395160865705, 19.35763547675022039249688490261, 19.89860833444156650714174888517, 21.04637918934778096985938879211

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