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Properties

Label 1-152-152.13-r1-0-0
Degree $1$
Conductor $152$
Sign $0.939 - 0.342i$
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

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

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(16.3346\)
Root analytic conductor: \(16.3346\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (1:\ ),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.713145089 - 0.4793137032i\)
\(L(\frac12)\) \(\approx\) \(2.713145089 - 0.4793137032i\)
\(L(1)\) \(\approx\) \(1.633711540 - 0.06160182119i\)
\(L(1)\) \(\approx\) \(1.633711540 - 0.06160182119i\)

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

−28.15529360911318373315241738550, −26.42930654182001983367311947802, −25.69075962741650383234234340930, −25.20372498215124013906214185054, −24.159489075717459295999744287887, −22.94295858517745021519261965078, −21.80843172931357187094734782889, −20.98958061525385310661111401155, −19.80853579525902514626652508930, −18.8232411034097911270827427492, −18.11218598912860510657260410596, −17.06855034162531849318056934345, −15.48850465850061810323144210347, −14.62776069224298655387441940249, −13.62489693048447661400912182634, −12.75361362492189646417073237036, −11.693748531779567528641780644178, −9.9174483207835625608600423141, −9.248126990928929224292835069822, −8.074258226345601572573025312, −6.57822015367083051346113287613, −5.99123086433924146019060666333, −3.940950110325439487471069435916, −2.495419225799202052965983856624, −1.61630031233527192578581419771, 1.05925045675331041707529437214, 2.818353948963394642029942825236, 3.90043193594689740609149016449, 5.28896605733965272389437792423, 6.58447461083810485946607667413, 8.12936108922611637288602566839, 9.15043171053177019643564403008, 10.0591785007803963167269697818, 10.949301603167037090269243925649, 12.7798077299231381333025857488, 13.84630367178264669127565008302, 14.19402595045357711344497886818, 15.942234186914606300646761635119, 16.44600840615761651164168788550, 17.66444700144758576566109422345, 18.94491529797465567099020192331, 20.15610587123674913376752161039, 20.64323730899230923624485495086, 21.77339040085331731588720801906, 22.55822136443239366214935205947, 23.97825437110676140293208963289, 25.152364329376125376775589230183, 25.67699376213321728293508459125, 26.74385676273061605993491124732, 27.52016933938536457250336002605

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