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Properties

Label 1-152-152.3-r0-0-0
Degree $1$
Conductor $152$
Sign $0.939 + 0.342i$
Analytic cond. $0.705885$
Root an. cond. $0.705885$
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  + (−0.766 + 0.642i)3-s + (0.939 + 0.342i)5-s + (0.5 − 0.866i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.939 + 0.342i)15-s + (0.173 + 0.984i)17-s + (0.173 + 0.984i)21-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (0.5 + 0.866i)27-s + (0.173 − 0.984i)29-s + (−0.5 + 0.866i)31-s + (0.939 + 0.342i)33-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (0.939 + 0.342i)5-s + (0.5 − 0.866i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.939 + 0.342i)15-s + (0.173 + 0.984i)17-s + (0.173 + 0.984i)21-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (0.5 + 0.866i)27-s + (0.173 − 0.984i)29-s + (−0.5 + 0.866i)31-s + (0.939 + 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(0.705885\)
Root analytic conductor: \(0.705885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (0:\ ),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.055147505 + 0.1864060495i\)
\(L(\frac12)\) \(\approx\) \(1.055147505 + 0.1864060495i\)
\(L(1)\) \(\approx\) \(1.016040444 + 0.1358480613i\)
\(L(1)\) \(\approx\) \(1.016040444 + 0.1358480613i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−28.1957002705350049759936910844, −27.36319087301648849678234291779, −25.43357646579803896040886745702, −25.23972368243243259962930295583, −24.10878348108735704814167569342, −23.10016002668160925277847231348, −22.1496282622801442055546535407, −21.1560776582423277091444010491, −20.22769884694170562923298471881, −18.46760720254314667788558236878, −18.16786597857419929851207903054, −17.21489665074555118764279310675, −16.09689431143308143477265265260, −14.87231525612902521910525597276, −13.4571380740557827278657830043, −12.77322316889226573880843227581, −11.709796018743117622799754356323, −10.59738966444526081038010503793, −9.35633146371332687757543838641, −8.082687565137970137906073786020, −6.801567673493227434481722663932, −5.5246668492328739591932217052, −5.01523283075603259631572093765, −2.56182821445394273248464401330, −1.39384694630014980632022902949, 1.35826066961898592121226886063, 3.351897306460218221524380147443, 4.65564642797347662385355872384, 5.85346096370221711351126001975, 6.72676504842274574946362819847, 8.42124383970182281649502516218, 9.749184982060516894429239579179, 10.743919453775954757615844173361, 11.24974639000448550676797496821, 12.95303399806794383769889631331, 13.93959438964057698167198352966, 14.967360619542880293889570131467, 16.34054943747229366822654152883, 17.03173368744640133312113236620, 17.926601995297490737826414523661, 18.94086360851583270383028807086, 20.63324783153667154578885809938, 21.28314100796764790397721358850, 21.99485432554366981882214614339, 23.31460453379460957541228723948, 23.83575279597829765484176495182, 25.272331692448009976733529273193, 26.54760678107885779679810786682, 26.80583408377264632888542499262, 28.289384049386586900868691916981

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