Properties

Degree 1
Conductor $ 3 \cdot 13 $
Sign $-0.289 - 0.957i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  i·2-s − 4-s i·5-s i·7-s + i·8-s − 10-s + i·11-s − 14-s + 16-s + 17-s + i·19-s + i·20-s + 22-s + 23-s − 25-s + ⋯
L(s,χ)  = 1  i·2-s − 4-s i·5-s i·7-s + i·8-s − 10-s + i·11-s − 14-s + 16-s + 17-s + i·19-s + i·20-s + 22-s + 23-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.289 - 0.957i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.289 - 0.957i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(39\)    =    \(3 \cdot 13\)
\( \varepsilon \)  =  $-0.289 - 0.957i$
motivic weight  =  \(0\)
character  :  $\chi_{39} (8, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 39,\ (0:\ ),\ -0.289 - 0.957i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4386508127 - 0.5911290131i$
$L(\frac12,\chi)$  $\approx$  $0.4386508127 - 0.5911290131i$
$L(\chi,1)$  $\approx$  0.7122768728 - 0.5510574790i
$L(1,\chi)$  $\approx$  0.7122768728 - 0.5510574790i

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

−34.9697847421190074010604808109, −34.651056027991960295507594805515, −33.46467213664354432008963661946, −32.13344917307688631793817645100, −31.17260490928020542770856151686, −29.86340765962051576826022577002, −28.15232678369304812452533630678, −26.96766031193091503352720295049, −25.93176082279167096021087802092, −24.89537064638670007828346503668, −23.63889220665247162934588253149, −22.33966088863493097516598013428, −21.52352525807990071262383287992, −19.01819307835590843948881653832, −18.42414958218211998992887923621, −16.91246141438668725059454690905, −15.47943734531682874780533871190, −14.60968086682799372342932810036, −13.24400853666377239958482369023, −11.37942738744729073125943998740, −9.57881399310781662543108732367, −8.152604763136015232421263379237, −6.63104821631916594538273241674, −5.413857892704111321495007632140, −3.19824738596972026817489788535, 1.44644689777198198232219669108, 3.80425591722021312730043264254, 5.1244197996221783615247081134, 7.724714662652271058725915988, 9.32880260094334591582083470883, 10.48385193142510416871340929007, 12.13958017258858580711967968647, 13.07909642048073899477993961213, 14.45696970989184107606019183111, 16.590997465949291998128448455411, 17.60309325995246723561931773593, 19.22986064998944432259266662671, 20.42217703480620667708780099436, 21.00715998712246994942864549557, 22.80873773430836750850693972587, 23.64912708879102660737052758840, 25.37117801756171483908803125651, 26.9286412089511274025083654824, 27.91621872906164513409521661355, 28.98514044179987437889329709641, 30.04694035434576463027424808216, 31.25690758691947740705946244949, 32.36178492354055562723227316033, 33.39250356719763671843716235868, 35.48979129935803396118850605518

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