Properties

Degree $1$
Conductor $709$
Sign $-0.641 - 0.766i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.769 − 0.638i)2-s + (0.949 − 0.314i)3-s + (0.185 + 0.982i)4-s + (−0.987 + 0.159i)5-s + (−0.931 − 0.364i)6-s + (−0.132 − 0.991i)7-s + (0.484 − 0.874i)8-s + (0.802 − 0.596i)9-s + (0.861 + 0.507i)10-s + (0.388 − 0.921i)11-s + (0.484 + 0.874i)12-s + (−0.132 + 0.991i)13-s + (−0.530 + 0.847i)14-s + (−0.887 + 0.461i)15-s + (−0.931 + 0.364i)16-s + (0.388 + 0.921i)17-s + ⋯
L(s,χ)  = 1  + (−0.769 − 0.638i)2-s + (0.949 − 0.314i)3-s + (0.185 + 0.982i)4-s + (−0.987 + 0.159i)5-s + (−0.931 − 0.364i)6-s + (−0.132 − 0.991i)7-s + (0.484 − 0.874i)8-s + (0.802 − 0.596i)9-s + (0.861 + 0.507i)10-s + (0.388 − 0.921i)11-s + (0.484 + 0.874i)12-s + (−0.132 + 0.991i)13-s + (−0.530 + 0.847i)14-s + (−0.887 + 0.461i)15-s + (−0.931 + 0.364i)16-s + (0.388 + 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.641 - 0.766i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.641 - 0.766i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.641 - 0.766i$
Motivic weight: \(0\)
Character: $\chi_{709} (82, \cdot )$
Sato-Tate group: $\mu(59)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.641 - 0.766i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.4348225923 - 0.9311221839i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.4348225923 - 0.9311221839i\)
\(L(\chi,1)\) \(\approx\) \(0.7364170048 - 0.4641859699i\)
\(L(1,\chi)\) \(\approx\) \(0.7364170048 - 0.4641859699i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.81270041338690082929556203314, −22.40358529761657000736213487753, −20.84776659745190188471386294502, −20.27544167328242064155383451397, −19.627670196211629546231522214377, −18.75766709623179785617806775961, −18.30275924287815303732837066513, −17.086988114027833650624697307194, −16.071245040318984654881080134214, −15.5285535482860317353847729011, −14.96064453819254385803854443712, −14.313609278066184003008088619851, −12.98723004227977874334124822038, −12.058489317119028173636945304908, −11.08826720132490354166910524410, −9.83590053872953872416887509406, −9.38629141989556687808946341407, −8.49931540610616207299364118414, −7.62571494496142571067448464282, −7.26685007643342748355677361828, −5.68095601502159324862383323937, −4.88431166221918401467085691832, −3.64230160695805771670621216042, −2.59091848440204449351158969437, −1.378342228112604562841607502776, 0.61383852779993615431088589294, 1.70445170039486317429031549735, 3.01316795513124424887567908694, 3.72194123008800020572199381017, 4.33736868690747787913867499856, 6.59700035153097102030590574509, 7.25846757561990192316592613284, 7.97277125101646157973128153232, 8.83037377432719783909416980082, 9.4851727559997904894512806219, 10.74331729635870810157075715084, 11.22461763462228575090289944483, 12.34429751492615955047213168972, 13.04535430067984753790149685450, 14.046188671421869157471963525631, 14.76491808157173917057910106935, 16.07763177006213651596737434206, 16.51151882095193057601376518385, 17.560097941457936106692315023471, 18.75007275173644982219747991529, 19.09269707384116510889728104494, 19.83402871564588213432477765568, 20.29677823483064774781335033606, 21.29001030546018431920556095464, 22.02506699295350188485461910663

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