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Properties

Label 2-1152-16.13-c1-0-13
Degree $2$
Conductor $1152$
Sign $0.757 + 0.653i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.27 − 1.27i)5-s − 0.158i·7-s + (3.79 − 3.79i)11-s + (4.21 + 4.21i)13-s − 3.05·17-s + (−2.15 − 2.15i)19-s − 2.82i·23-s + 1.76i·25-s + (2.09 + 2.09i)29-s − 4.15·31-s + (−0.202 − 0.202i)35-s + (5.98 − 5.98i)37-s − 2.60i·41-s + (5.75 − 5.75i)43-s − 2.82·47-s + ⋯
L(s)  = 1  + (0.568 − 0.568i)5-s − 0.0600i·7-s + (1.14 − 1.14i)11-s + (1.16 + 1.16i)13-s − 0.740·17-s + (−0.495 − 0.495i)19-s − 0.589i·23-s + 0.353i·25-s + (0.389 + 0.389i)29-s − 0.746·31-s + (−0.0341 − 0.0341i)35-s + (0.984 − 0.984i)37-s − 0.406i·41-s + (0.877 − 0.877i)43-s − 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.757 + 0.653i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.757 + 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.954019831\)
\(L(\frac12)\) \(\approx\) \(1.954019831\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-1.27 + 1.27i)T - 5iT^{2} \)
7 \( 1 + 0.158iT - 7T^{2} \)
11 \( 1 + (-3.79 + 3.79i)T - 11iT^{2} \)
13 \( 1 + (-4.21 - 4.21i)T + 13iT^{2} \)
17 \( 1 + 3.05T + 17T^{2} \)
19 \( 1 + (2.15 + 2.15i)T + 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (-2.09 - 2.09i)T + 29iT^{2} \)
31 \( 1 + 4.15T + 31T^{2} \)
37 \( 1 + (-5.98 + 5.98i)T - 37iT^{2} \)
41 \( 1 + 2.60iT - 41T^{2} \)
43 \( 1 + (-5.75 + 5.75i)T - 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (-3.55 + 3.55i)T - 53iT^{2} \)
59 \( 1 + (4 - 4i)T - 59iT^{2} \)
61 \( 1 + (3.66 + 3.66i)T + 61iT^{2} \)
67 \( 1 + (-0.767 - 0.767i)T + 67iT^{2} \)
71 \( 1 + 0.317iT - 71T^{2} \)
73 \( 1 - 1.33iT - 73T^{2} \)
79 \( 1 - 9.69T + 79T^{2} \)
83 \( 1 + (0.115 + 0.115i)T + 83iT^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + 0.571T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.219172782856626451711334234547, −9.094699989510457135688956135952, −8.399888304026330515510494738994, −6.99303375124325256222142254107, −6.32604237750801891687825335406, −5.60669306153293526584838384314, −4.36196111504773499221150372404, −3.67880576484288635016962254215, −2.14163664663664782010807357065, −0.997252152018289728472507164930, 1.38580784535627046025192429030, 2.54678918931375897201650328347, 3.72819861477195252743292953035, 4.62712912396091530873137679414, 5.98918906986356418278015658041, 6.34742267793101433786463625577, 7.36263027631526199523235205070, 8.273827107360188888101989604245, 9.181591890857744269363114246181, 9.908061140295156459774469313942

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