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Properties

Label 1-139-139.124-r0-0-0
Degree $1$
Conductor $139$
Sign $-0.103 + 0.994i$
Analytic cond. $0.645513$
Root an. cond. $0.645513$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $-0.103 + 0.994i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ -0.103 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6300366774 + 0.6993078987i\)
\(L(\frac12)\) \(\approx\) \(0.6300366774 + 0.6993078987i\)
\(L(1)\) \(\approx\) \(0.7637694890 + 0.4946288936i\)
\(L(1)\) \(\approx\) \(0.7637694890 + 0.4946288936i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (-0.877 + 0.480i)T \)
3 \( 1 + (0.113 + 0.993i)T \)
5 \( 1 + (0.613 + 0.789i)T \)
7 \( 1 + (0.995 + 0.0909i)T \)
11 \( 1 + (0.934 - 0.356i)T \)
13 \( 1 + (0.803 - 0.595i)T \)
17 \( 1 + (0.983 - 0.181i)T \)
19 \( 1 + (-0.648 - 0.761i)T \)
23 \( 1 + (-0.576 + 0.816i)T \)
29 \( 1 + (-0.998 - 0.0455i)T \)
31 \( 1 + (-0.974 - 0.225i)T \)
37 \( 1 + (-0.715 - 0.699i)T \)
41 \( 1 + (-0.949 + 0.313i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.377 + 0.926i)T \)
53 \( 1 + (0.0227 - 0.999i)T \)
59 \( 1 + (0.460 - 0.887i)T \)
61 \( 1 + (-0.949 - 0.313i)T \)
67 \( 1 + (-0.158 - 0.987i)T \)
71 \( 1 + (0.377 - 0.926i)T \)
73 \( 1 + (-0.829 - 0.557i)T \)
79 \( 1 + (0.203 + 0.979i)T \)
83 \( 1 + (-0.158 + 0.987i)T \)
89 \( 1 + (-0.247 - 0.968i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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