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Properties

Label 1-152-152.85-r0-0-0
Degree $1$
Conductor $152$
Sign $0.189 - 0.981i$
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  + (−0.173 + 0.984i)3-s + (−0.766 − 0.642i)5-s + (−0.5 − 0.866i)7-s + (−0.939 − 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.173 − 0.984i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)17-s + (0.939 − 0.342i)21-s + (0.766 − 0.642i)23-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + (0.766 + 0.642i)33-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)3-s + (−0.766 − 0.642i)5-s + (−0.5 − 0.866i)7-s + (−0.939 − 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.173 − 0.984i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)17-s + (0.939 − 0.342i)21-s + (0.766 − 0.642i)23-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + (0.766 + 0.642i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(0.705885\)
Root analytic conductor: \(0.705885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (0:\ ),\ 0.189 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4755414626 - 0.3926645631i\)
\(L(\frac12)\) \(\approx\) \(0.4755414626 - 0.3926645631i\)
\(L(1)\) \(\approx\) \(0.7281533424 - 0.09717988080i\)
\(L(1)\) \(\approx\) \(0.7281533424 - 0.09717988080i\)

Euler product

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

−28.45551245739896540896127207657, −27.33448815068203794869144052952, −26.141302166260291557634822631483, −25.25109524110191807783416300290, −24.369509965157953134141842636294, −23.23270504955660572281097948544, −22.63964268498952998541616663838, −21.59644415351402969514762508237, −19.840174822907858379724723285881, −19.3367828124662576782092796609, −18.3962890803164148677812908055, −17.546298570186823134561718840381, −16.16587067006904478617550705076, −15.10428742737519924268540444212, −14.11438925440100833437958673271, −12.7988576626255694248367357882, −11.9111860354764694560205927067, −11.22037034698867660243572841135, −9.50199874533890314945092416291, −8.35398589818762108004664914919, −6.96081835309090085523492526055, −6.550549404150990649572365043840, −4.823772771372830919986884356260, −3.13724848444097651191787489457, −1.916819940325073321201091287780, 0.53473197271191062544704597079, 3.25349320694351402246867724250, 4.1076645529400585552660396747, 5.23599513391530908065377394368, 6.67440362106723135474269986441, 8.22302601356622551635883585195, 9.10773792314473602784116738016, 10.43391640450616824683667322156, 11.18975863640562940819504862148, 12.46491476456638162101520893150, 13.62598661979095961102777328652, 14.96277074992087900623518494967, 15.874728479471387847193015282447, 16.67712099900143812189146856309, 17.4338029072259171164107898206, 19.25118202068218630195605838550, 20.03984607074496185871849038949, 20.74083265477117660672882456916, 22.06506182536772606374072391331, 22.8061434082109057420170791638, 23.77607854768073803841196551332, 24.84393717699226110741986280418, 26.19292309045443779993979585126, 27.02258179751988437895242809024, 27.568933654966626941105039063342

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