Properties

Degree 1
Conductor 79
Sign $-0.582 - 0.813i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.845 + 0.534i)2-s + (−0.799 − 0.600i)3-s + (0.428 − 0.903i)4-s + (−0.632 + 0.774i)5-s + (0.996 + 0.0804i)6-s + (0.919 − 0.391i)7-s + (0.120 + 0.992i)8-s + (0.278 + 0.960i)9-s + (0.120 − 0.992i)10-s + (0.987 − 0.160i)11-s + (−0.885 + 0.464i)12-s + (−0.996 + 0.0804i)13-s + (−0.568 + 0.822i)14-s + (0.970 − 0.239i)15-s + (−0.632 − 0.774i)16-s + (−0.568 − 0.822i)17-s + ⋯
L(s,χ)  = 1  + (−0.845 + 0.534i)2-s + (−0.799 − 0.600i)3-s + (0.428 − 0.903i)4-s + (−0.632 + 0.774i)5-s + (0.996 + 0.0804i)6-s + (0.919 − 0.391i)7-s + (0.120 + 0.992i)8-s + (0.278 + 0.960i)9-s + (0.120 − 0.992i)10-s + (0.987 − 0.160i)11-s + (−0.885 + 0.464i)12-s + (−0.996 + 0.0804i)13-s + (−0.568 + 0.822i)14-s + (0.970 − 0.239i)15-s + (−0.632 − 0.774i)16-s + (−0.568 − 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.582 - 0.813i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.582 - 0.813i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $-0.582 - 0.813i$
motivic weight  =  \(0\)
character  :  $\chi_{79} (75, \cdot )$
Sato-Tate  :  $\mu(78)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 79,\ (1:\ ),\ -0.582 - 0.813i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1241111287 - 0.2414528542i$
$L(\frac12,\chi)$  $\approx$  $0.1241111287 - 0.2414528542i$
$L(\chi,1)$  $\approx$  0.4664573875 + 0.009403598448i
$L(1,\chi)$  $\approx$  0.4664573875 + 0.009403598448i

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

−31.07284669880009183869203185811, −29.945472541533847670507168272742, −28.75992438840075620201968591930, −27.72789643449515306869649153240, −27.56529862577906024775242245854, −26.350248631451303666071035282527, −24.713694426697792429087638951228, −23.86754736642126702448722902527, −22.15915542228263297295023242581, −21.46262994673260531134520521298, −20.24406782593541825575191991375, −19.38245424999392041534966528043, −17.63859502956277754240968470219, −17.25214435479758072236577411321, −16.001481838862194736928606729081, −14.95849677553143576530010146771, −12.58542591874311658854298243218, −11.78241985748283836854195853851, −10.95919765159156498641566761618, −9.460636301244584612131699878065, −8.54124865864756755110971833464, −7.02300187048061535414603088733, −5.04733528469081663098721235517, −3.8748454066539234458272605278, −1.53117053412417794153845511859, 0.19652122373473538458009893951, 1.93866304663944842706045396405, 4.62658929211868552601029304570, 6.276696465490354138883864152538, 7.25156487109556241842816405914, 8.13048353132095166460521321809, 10.02068028991111117960626073367, 11.22790040288752140420378647083, 11.86678593077653080726098288554, 14.058288882320072759980699535319, 14.93773535405066222392010816946, 16.46003461020083037111837077525, 17.29896922693703728178048423416, 18.31320185749774770278795656300, 19.14210014479516798678604783750, 20.26309311651867595840271238784, 22.23950853941071291954777741515, 23.12557098868949825150646479123, 24.291414515284288709805571331073, 24.81279252017890917980613404296, 26.55263972966741507791602424620, 27.25708994376828821825256951552, 28.060508423881909817053637720591, 29.54950008259297305077006894177, 30.05036177142924100504510376521

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