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Properties

Label 1-152-152.21-r1-0-0
Degree $1$
Conductor $152$
Sign $-0.660 - 0.750i$
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  + (0.173 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.660 - 0.750i$
Analytic conductor: \(16.3346\)
Root analytic conductor: \(16.3346\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (1:\ ),\ -0.660 - 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2207430146 + 0.4880668175i\)
\(L(\frac12)\) \(\approx\) \(-0.2207430146 + 0.4880668175i\)
\(L(1)\) \(\approx\) \(0.6262104263 + 0.4481592846i\)
\(L(1)\) \(\approx\) \(0.6262104263 + 0.4481592846i\)

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

−26.933874515209393049268467666007, −26.33602358627598366301630959081, −24.96840402057042753168882793889, −24.15191826040648003122548017045, −23.50752376565196305217762889630, −22.55828195345812574381747273153, −21.049496673256375717457508034683, −19.81942569296026117138863853544, −19.47546678165732544306944283734, −18.40177156860276829171044860813, −16.91233296505366922941263436610, −16.47669088103824302987544964242, −14.93639355940776181645502737062, −13.671738518833893820775199994126, −13.04656443117579090510264364436, −11.84251576105253753702600043630, −11.01274304198024016388622199333, −9.1566768012480019349785614671, −8.3392273395976676385768055224, −7.1162019789863758075375972098, −6.27856930735075838576929804132, −4.44762275980022318510672575924, −3.2816430053983295711963507286, −1.418594330787170482250079482240, −0.20311139114352733988174737869, 2.57720688632008346454079489720, 3.57862928482130973992215785694, 4.80942690944565087095190113509, 6.17447980300980093044851968353, 7.57444154341208162890414189335, 8.87245465577030323840240387067, 9.790778197939737818336999495429, 10.97483261271692886198757635655, 11.85110897977586256544510480094, 13.18329958978107399329033505401, 14.84136584416739702933540063013, 15.22131119571571491878242898346, 16.05296765404217000150808687661, 17.40056559799335722111630012462, 18.558463756444664455527172659092, 19.70648259853464786244758715961, 20.3685315558306860036816233066, 21.746553781681601113163508839073, 22.531429240810492031189178218916, 23.043974382289153201417789703709, 24.75873684077251374309078693369, 25.65136759868153316305584414592, 26.48067915427094378769281398126, 27.60501576020138083799554094428, 27.978720525181477296255675022852

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