Properties

Degree $1$
Conductor $77$
Sign $-0.586 + 0.810i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.669 + 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.978 + 0.207i)5-s + (−0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (−0.104 + 0.994i)18-s + (0.104 + 0.994i)19-s + (−0.309 + 0.951i)20-s + ⋯
L(s,χ)  = 1  + (0.669 + 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.978 + 0.207i)5-s + (−0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (−0.104 + 0.994i)18-s + (0.104 + 0.994i)19-s + (−0.309 + 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.586 + 0.810i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.586 + 0.810i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.586 + 0.810i$
Motivic weight: \(0\)
Character: $\chi_{77} (3, \cdot )$
Sato-Tate group: $\mu(30)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ -0.586 + 0.810i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.7736819763 + 1.515170026i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.7736819763 + 1.515170026i\)
\(L(\chi,1)\) \(\approx\) \(1.030311494 + 0.6985932746i\)
\(L(1,\chi)\) \(\approx\) \(1.030311494 + 0.6985932746i\)

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

−30.28835878872832631820884771782, −29.52399610904276027003691332126, −28.63297288327858779828651374088, −27.88508899077680048093039937145, −26.61195572979658409140817564860, −24.86554865242034349131960103755, −23.95568963126336299944382854255, −22.5522483830530171308375491073, −22.08567227360948118599429646848, −20.97670086100806126100640763318, −20.06902473950750367380198552490, −18.30643353840229513972075901681, −17.543749869340090550842111573288, −16.06715639868272273161481238065, −14.83356356190901978999292362670, −13.42258435737894783913978157172, −12.51126769771068909210301141006, −11.21821181350734695983081027819, −10.22660219465915206680515992916, −9.25593789725965684167288585463, −6.602691621845375806366524676119, −5.47333628164352576747290721417, −4.54048142083980698752273476317, −2.62832271304555309694413566228, −0.753916284463055305237652155096, 2.04676806342713841879522778570, 4.27800707114344239702214413683, 5.71890058097564747840369072277, 6.41946855837561080809690435167, 7.70705969479018772753325266006, 9.525448668566125974771280350199, 11.14061951266124812113497948716, 12.39367156841019988846864067431, 13.42080415541261533065836803401, 14.411484468662293765018620342901, 15.93164068988560249262885086824, 17.05771220837937454437859394755, 17.6769343379869520300524376266, 18.93804745653302096800213506302, 21.022330514463974442120039584094, 21.89318252843437468427624167929, 22.69873433469055811505420842384, 23.90101135386405786312603327668, 24.6481088670779487579369966453, 25.75679354130628031942655045326, 26.843902411413797602432095745695, 28.41928160929695278199624036435, 29.43325120203105852652235515440, 30.233367495234399647512353465940, 31.41662912914344865145375976788

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