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Properties

Label 1-153-153.29-r0-0-0
Degree $1$
Conductor $153$
Sign $-0.0648 - 0.997i$
Analytic cond. $0.710529$
Root an. cond. $0.710529$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (0.793 − 0.608i)5-s + (−0.793 − 0.608i)7-s + (−0.707 − 0.707i)8-s + (−0.923 + 0.382i)10-s + (0.608 − 0.793i)11-s + (−0.866 − 0.5i)13-s + (0.608 + 0.793i)14-s + (0.5 + 0.866i)16-s + (−0.707 + 0.707i)19-s + (0.991 − 0.130i)20-s + (−0.793 + 0.608i)22-s + (0.991 + 0.130i)23-s + (0.258 − 0.965i)25-s + (0.707 + 0.707i)26-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (0.793 − 0.608i)5-s + (−0.793 − 0.608i)7-s + (−0.707 − 0.707i)8-s + (−0.923 + 0.382i)10-s + (0.608 − 0.793i)11-s + (−0.866 − 0.5i)13-s + (0.608 + 0.793i)14-s + (0.5 + 0.866i)16-s + (−0.707 + 0.707i)19-s + (0.991 − 0.130i)20-s + (−0.793 + 0.608i)22-s + (0.991 + 0.130i)23-s + (0.258 − 0.965i)25-s + (0.707 + 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0648 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0648 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.0648 - 0.997i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ -0.0648 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4733416763 - 0.5051213206i\)
\(L(\frac12)\) \(\approx\) \(0.4733416763 - 0.5051213206i\)
\(L(1)\) \(\approx\) \(0.6596726469 - 0.2829276906i\)
\(L(1)\) \(\approx\) \(0.6596726469 - 0.2829276906i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (0.793 - 0.608i)T \)
7 \( 1 + (-0.793 - 0.608i)T \)
11 \( 1 + (0.608 - 0.793i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (0.991 + 0.130i)T \)
29 \( 1 + (-0.130 - 0.991i)T \)
31 \( 1 + (-0.608 - 0.793i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (0.130 - 0.991i)T \)
43 \( 1 + (-0.258 + 0.965i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (0.965 - 0.258i)T \)
61 \( 1 + (0.793 + 0.608i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (-0.608 + 0.793i)T \)
83 \( 1 + (0.965 + 0.258i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.130 + 0.991i)T \)
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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

−28.2983363085972893509775927888, −27.200385587815728262454037794044, −26.18722086092441727568836060318, −25.429645775001970857526671905700, −24.870318629327544328735216781930, −23.49506079057024709871017649572, −22.23148107244289102336652517010, −21.442354724406955368030261173507, −20.02224009040918868005972034823, −19.18719905943312271566788944070, −18.297168996997269528193534018555, −17.33574168598487044280410531839, −16.53524921626397186218560010930, −15.13414144496788405279560956946, −14.57626689956884095318764582501, −12.948699273959704687923049743926, −11.74157395225671466758547883441, −10.466852647349191782918780286953, −9.55216234924711091447155296532, −8.87375898400083409346308302483, −7.00968472379296018586351823026, −6.60257492888652709222479672932, −5.17938135554434298838974155252, −2.91991761253357436684103787248, −1.86041267007391990992944467691, 0.80370347766917070517475400664, 2.372951156810953749663136824518, 3.816672108661990309520504864921, 5.748139167586603808830219737802, 6.80744940179248992482472272792, 8.12713939369360895666564762589, 9.308407713087548296676348212076, 9.967350712088731272093359335403, 11.09572788937731289259150138167, 12.48748954306872550471010266837, 13.26185309784672800277135524521, 14.718648877340411326043728513635, 16.246311335533586482720983648586, 16.88269529814398042552797143506, 17.57638673380463180680156604948, 18.99947121412886321260646990602, 19.67771565353946293742084862724, 20.71122343858037222049271880764, 21.57749867532898191591078031669, 22.690839280437872971721188381747, 24.27712266080538865360052098470, 25.00160104427069152600399508857, 25.85731311879367017827981578208, 26.88161242010420347316989669261, 27.67634494624341571894471782329

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