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Properties

Label 1-152-152.139-r1-0-0
Degree $1$
Conductor $152$
Sign $0.188 + 0.982i$
Analytic cond. $16.3346$
Root an. cond. $16.3346$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.188 + 0.982i$
Analytic conductor: \(16.3346\)
Root analytic conductor: \(16.3346\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (1:\ ),\ 0.188 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.108265415 + 1.741996448i\)
\(L(\frac12)\) \(\approx\) \(2.108265415 + 1.741996448i\)
\(L(1)\) \(\approx\) \(1.523674609 + 0.5926414312i\)
\(L(1)\) \(\approx\) \(1.523674609 + 0.5926414312i\)

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

−27.29345364466320196428494035296, −26.392770303754277346737279018029, −25.73097724327480153015620130257, −24.54878199868584310517699050216, −24.00846006443220727677551641433, −22.72094797143233212391991552486, −21.40175741580903544402445343017, −20.720458877315422256589886997283, −19.60153466744318311059448834040, −18.65450968289720393135935399666, −17.642419514448354984315620568915, −16.87245340845466342631140238470, −15.09985640223512568448076854059, −14.28823908812190882616536292800, −13.4514960149497606701754771382, −12.639174669099420333655379988450, −10.92785470272299273040899470385, −10.045546017381901717558589394236, −8.69698169579601400481941378432, −7.67039629931103945059416793398, −6.624259743032591141027276594348, −5.30324374625113480792810330429, −3.50997031172828984595356144347, −2.33160263487458275828304517618, −0.97644975791048849645702335794, 1.89702399862915952820087819117, 2.754466858056584275864221377113, 4.700118545513268731546035192350, 5.30160910961958215889537137375, 7.117633934747334405250310958027, 8.47678063571413626698646304389, 9.423805803464026876053100471560, 10.09274321873780669644594459508, 11.64249385500772211338003031012, 12.90044360477577947369119267444, 13.99642085351810343176393037323, 14.84777667421419938400762546000, 15.79501018725553368787755033276, 16.99441559280151306214310292014, 18.038278131787555233638795503963, 19.12364147894802215531740687835, 20.381368031777013941001768287287, 21.170750241741520075179537638118, 21.71120384236465473669995796546, 22.93845640370368693683125935348, 24.63878147086585182799038056535, 24.98755854146173464891803267319, 26.00229065785450179969271602649, 26.950207413550705520040023350934, 28.050702265563405953753304797472

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