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Properties

Label 1-139-139.28-r0-0-0
Degree $1$
Conductor $139$
Sign $0.266 + 0.963i$
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  + (−0.715 + 0.699i)2-s + (−0.949 + 0.313i)3-s + (0.0227 − 0.999i)4-s + (−0.829 + 0.557i)5-s + (0.460 − 0.887i)6-s + (0.538 − 0.842i)7-s + (0.682 + 0.730i)8-s + (0.803 − 0.595i)9-s + (0.203 − 0.979i)10-s + (−0.648 − 0.761i)11-s + (0.291 + 0.956i)12-s + (0.746 + 0.665i)13-s + (0.203 + 0.979i)14-s + (0.613 − 0.789i)15-s + (−0.998 − 0.0455i)16-s + (−0.419 + 0.907i)17-s + ⋯
L(s)  = 1  + (−0.715 + 0.699i)2-s + (−0.949 + 0.313i)3-s + (0.0227 − 0.999i)4-s + (−0.829 + 0.557i)5-s + (0.460 − 0.887i)6-s + (0.538 − 0.842i)7-s + (0.682 + 0.730i)8-s + (0.803 − 0.595i)9-s + (0.203 − 0.979i)10-s + (−0.648 − 0.761i)11-s + (0.291 + 0.956i)12-s + (0.746 + 0.665i)13-s + (0.203 + 0.979i)14-s + (0.613 − 0.789i)15-s + (−0.998 − 0.0455i)16-s + (−0.419 + 0.907i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ 0.266 + 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3926325998 + 0.2989332113i\)
\(L(\frac12)\) \(\approx\) \(0.3926325998 + 0.2989332113i\)
\(L(1)\) \(\approx\) \(0.5006682750 + 0.2159046924i\)
\(L(1)\) \(\approx\) \(0.5006682750 + 0.2159046924i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (-0.715 + 0.699i)T \)
3 \( 1 + (-0.949 + 0.313i)T \)
5 \( 1 + (-0.829 + 0.557i)T \)
7 \( 1 + (0.538 - 0.842i)T \)
11 \( 1 + (-0.648 - 0.761i)T \)
13 \( 1 + (0.746 + 0.665i)T \)
17 \( 1 + (-0.419 + 0.907i)T \)
19 \( 1 + (0.995 - 0.0909i)T \)
23 \( 1 + (0.460 + 0.887i)T \)
29 \( 1 + (-0.877 + 0.480i)T \)
31 \( 1 + (0.803 + 0.595i)T \)
37 \( 1 + (0.613 + 0.789i)T \)
41 \( 1 + (0.934 + 0.356i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.898 - 0.439i)T \)
53 \( 1 + (-0.247 - 0.968i)T \)
59 \( 1 + (0.854 - 0.519i)T \)
61 \( 1 + (0.934 - 0.356i)T \)
67 \( 1 + (0.983 + 0.181i)T \)
71 \( 1 + (0.898 + 0.439i)T \)
73 \( 1 + (-0.974 + 0.225i)T \)
79 \( 1 + (-0.775 - 0.631i)T \)
83 \( 1 + (0.983 - 0.181i)T \)
89 \( 1 + (0.377 + 0.926i)T \)
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

−28.41985559762873050832409841092, −27.55878467925585457704317096799, −26.695204519022252007862545053001, −25.1145204403167939108023126537, −24.39640237063934127903582623908, −23.01297158703014072966816369092, −22.35685769868779671672146898824, −20.90650689006322915953677642267, −20.3428228285492745611799823493, −18.84064431727689343540501311413, −18.26082976855988266476913120754, −17.38183926744203526322261072192, −16.12154188302457667592881642150, −15.4844765966638476442728453879, −13.2170466171239717832786242071, −12.36201117496472124328016467027, −11.59500898975677980319935168454, −10.780101455629488572907430989, −9.35639715344104085408294055493, −8.10318105477882597821325629609, −7.29168463427666988067308821005, −5.435345373657925106370380553382, −4.31569808996040722932205060813, −2.44297034265941140857610090548, −0.8313378602064466591976290422, 1.105820812201600123018884597810, 3.79200818067997454782863530689, 5.07389419526349721576116404566, 6.37153097954096587126150345115, 7.33086087359088366348457708922, 8.38091041245691227992279654709, 9.95192767947868696526635996953, 11.10674963164653268678721381840, 11.30511840423406715593982659534, 13.40272726873586336384510411471, 14.668045542839781827300645858194, 15.76385888393522402465967441479, 16.37730383057769849068244084235, 17.47646945670218054763550076223, 18.34675718483800038555994173667, 19.24808553586497012101096434210, 20.5245482593925927845056085579, 21.82976916562779360585182673776, 23.17974455001259625609045482122, 23.61652195994164701942720388195, 24.35350671386794776874627358300, 26.22445983658431388513568502799, 26.62494501814743922427003805692, 27.46446097316493610414320393426, 28.381404570862005379176697260422

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