Properties

Degree $1$
Conductor $709$
Sign $0.309 + 0.950i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.185 − 0.982i)2-s + (0.802 + 0.596i)3-s + (−0.931 − 0.364i)4-s + (0.949 + 0.314i)5-s + (0.734 − 0.678i)6-s + (−0.964 − 0.263i)7-s + (−0.530 + 0.847i)8-s + (0.288 + 0.957i)9-s + (0.484 − 0.874i)10-s + (−0.697 + 0.716i)11-s + (−0.530 − 0.847i)12-s + (−0.964 + 0.263i)13-s + (−0.437 + 0.899i)14-s + (0.574 + 0.818i)15-s + (0.734 + 0.678i)16-s + (−0.697 − 0.716i)17-s + ⋯
L(s,χ)  = 1  + (0.185 − 0.982i)2-s + (0.802 + 0.596i)3-s + (−0.931 − 0.364i)4-s + (0.949 + 0.314i)5-s + (0.734 − 0.678i)6-s + (−0.964 − 0.263i)7-s + (−0.530 + 0.847i)8-s + (0.288 + 0.957i)9-s + (0.484 − 0.874i)10-s + (−0.697 + 0.716i)11-s + (−0.530 − 0.847i)12-s + (−0.964 + 0.263i)13-s + (−0.437 + 0.899i)14-s + (0.574 + 0.818i)15-s + (0.734 + 0.678i)16-s + (−0.697 − 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.309 + 0.950i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.309 + 0.950i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.309 + 0.950i$
Motivic weight: \(0\)
Character: $\chi_{709} (339, \cdot )$
Sato-Tate group: $\mu(59)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.309 + 0.950i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.9581882529 + 0.6954923743i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.9581882529 + 0.6954923743i\)
\(L(\chi,1)\) \(\approx\) \(1.135488921 - 0.03462633493i\)
\(L(1,\chi)\) \(\approx\) \(1.135488921 - 0.03462633493i\)

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

−22.4163825108195779441849691509, −21.737591587973482251435451514476, −21.051017557395621311708971011230, −19.77548634909010417389795962066, −19.1654347312349312475819532515, −18.20036010496885789997063942483, −17.576407124535938843354281736306, −16.71624775243837406785786434083, −15.752142362133812254817211118537, −15.063020545233167930513363170804, −14.1091652252706079842170512374, −13.347951121585042465776308977089, −12.95952872057688032261876704404, −12.1752973856250913147911631718, −10.248527394648129392406073705667, −9.60281730716273745620399363796, −8.64454731075469441915026429609, −8.18132181096559514555881057477, −6.83710624040552757866116295057, −6.37008612429861978314631240479, −5.449036491608284961848372953454, −4.29550044201165589168569184376, −2.99855134554455960695557763985, −2.27013705867245308667055485483, −0.43529889050232892088510405320, 1.8143478291746799211783121303, 2.54548712860799419690512282828, 3.26407920524401133369650439186, 4.43250845885288685873692172299, 5.16245434916615464947745599193, 6.423570824145987013961319341203, 7.60312223005525206903602962101, 8.86370622575457411552445277437, 9.64383940309339056588156778703, 10.12269259929942298648896124351, 10.65178734839891163202268884112, 12.1346862248038524076808186458, 12.908338790061342240839723588280, 13.74246044621666706966966232683, 14.20828746255783679428135789446, 15.17896682037656953666634948137, 16.11302994289296020858104554784, 17.244685804242651556263069818763, 18.0687112396510263832901860911, 18.970583126111691786631475477, 19.75254183198300373962710590015, 20.292104329519254968482733986, 21.29799165391338013883786404176, 21.63883161648256452903872776724, 22.595970753499484001007427018700

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