Properties

Degree 1
Conductor 139
Sign $-0.103 - 0.994i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.877 − 0.480i)2-s + (0.113 − 0.993i)3-s + (0.538 + 0.842i)4-s + (0.613 − 0.789i)5-s + (−0.576 + 0.816i)6-s + (0.995 − 0.0909i)7-s + (−0.0682 − 0.997i)8-s + (−0.974 − 0.225i)9-s + (−0.917 + 0.398i)10-s + (0.934 + 0.356i)11-s + (0.898 − 0.439i)12-s + (0.803 + 0.595i)13-s + (−0.917 − 0.398i)14-s + (−0.715 − 0.699i)15-s + (−0.419 + 0.907i)16-s + (0.983 + 0.181i)17-s + ⋯
L(s,χ)  = 1  + (−0.877 − 0.480i)2-s + (0.113 − 0.993i)3-s + (0.538 + 0.842i)4-s + (0.613 − 0.789i)5-s + (−0.576 + 0.816i)6-s + (0.995 − 0.0909i)7-s + (−0.0682 − 0.997i)8-s + (−0.974 − 0.225i)9-s + (−0.917 + 0.398i)10-s + (0.934 + 0.356i)11-s + (0.898 − 0.439i)12-s + (0.803 + 0.595i)13-s + (−0.917 − 0.398i)14-s + (−0.715 − 0.699i)15-s + (−0.419 + 0.907i)16-s + (0.983 + 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.103 - 0.994i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.103 - 0.994i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(139\)
\( \varepsilon \)  =  $-0.103 - 0.994i$
motivic weight  =  \(0\)
character  :  $\chi_{139} (37, \cdot )$
Sato-Tate  :  $\mu(69)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 139,\ (0:\ ),\ -0.103 - 0.994i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6300366774 - 0.6993078987i$
$L(\frac12,\chi)$  $\approx$  $0.6300366774 - 0.6993078987i$
$L(\chi,1)$  $\approx$  0.7637694890 - 0.4946288936i
$L(1,\chi)$  $\approx$  0.7637694890 - 0.4946288936i

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

−28.091118551547266390570630838221, −27.654833463912051711026607190218, −26.78362394083396898366622024085, −25.74064657883105807515067387655, −25.24139407916324628222839626276, −23.89042905532081763275053751800, −22.67913294746485493593470704894, −21.557172097625323672737858949043, −20.722280953116012112182729596786, −19.59565672300931148403701158776, −18.37722939322873716507673915809, −17.49435082947609205295397276280, −16.67949472633042270638405353460, −15.39414978321204384912707452770, −14.65261602621927317131428491997, −13.8988225546395480875135470754, −11.39434685560463612541628831739, −10.87511128770597042800412501347, −9.74820199114927537189107850745, −8.8351569720366528126550027481, −7.69422627050140248209770876675, −6.16214777276610483370272480935, −5.27221248110168882517862466777, −3.44911216370210937659139722554, −1.7908198802445061052479022593, 1.39437634324382771595631473585, 1.8859825407557914143535306494, 3.92380014235499336413789144961, 5.8304584287769823497751195710, 7.09758854276105651901002903798, 8.36166899183045692720004450193, 8.907897008112983722345197226940, 10.37178844580392554507098859183, 11.755727706922988782126596849808, 12.35165762758737360603023196656, 13.5865134183313445814256735126, 14.651895092787986995618305394555, 16.68990684703635477727721670442, 17.08762568597554962584678302909, 18.212309577313402207885336188566, 18.88620736238965121121379322239, 20.27556288036294509886448927225, 20.68608753398434242482870714488, 21.872096026737034170611414178637, 23.5167355015166372379908294290, 24.48795361725824360154740666087, 25.25251541701432763510665250341, 25.98665462411568490274938321661, 27.55132581442466061407856252516, 28.12859176977807723428427658754

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