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Properties

Label 2-1152-48.35-c3-0-25
Degree $2$
Conductor $1152$
Sign $-0.135 - 0.990i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (15.2 + 15.2i)5-s + 24.4·7-s + (−20.1 + 20.1i)11-s + (26.7 + 26.7i)13-s − 85.9i·17-s + (−53.2 + 53.2i)19-s + 119. i·23-s + 337. i·25-s + (78.5 − 78.5i)29-s + 200. i·31-s + (372. + 372. i)35-s + (76.9 − 76.9i)37-s + 279.·41-s + (−15.7 − 15.7i)43-s − 470.·47-s + ⋯
L(s)  = 1  + (1.36 + 1.36i)5-s + 1.32·7-s + (−0.553 + 0.553i)11-s + (0.571 + 0.571i)13-s − 1.22i·17-s + (−0.643 + 0.643i)19-s + 1.08i·23-s + 2.69i·25-s + (0.502 − 0.502i)29-s + 1.16i·31-s + (1.79 + 1.79i)35-s + (0.342 − 0.342i)37-s + 1.06·41-s + (−0.0559 − 0.0559i)43-s − 1.46·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.135 - 0.990i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ -0.135 - 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.177966622\)
\(L(\frac12)\) \(\approx\) \(3.177966622\)
\(L(\frac{5}{2})\) 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 \( 1 + (-15.2 - 15.2i)T + 125iT^{2} \)
7 \( 1 - 24.4T + 343T^{2} \)
11 \( 1 + (20.1 - 20.1i)T - 1.33e3iT^{2} \)
13 \( 1 + (-26.7 - 26.7i)T + 2.19e3iT^{2} \)
17 \( 1 + 85.9iT - 4.91e3T^{2} \)
19 \( 1 + (53.2 - 53.2i)T - 6.85e3iT^{2} \)
23 \( 1 - 119. iT - 1.21e4T^{2} \)
29 \( 1 + (-78.5 + 78.5i)T - 2.43e4iT^{2} \)
31 \( 1 - 200. iT - 2.97e4T^{2} \)
37 \( 1 + (-76.9 + 76.9i)T - 5.06e4iT^{2} \)
41 \( 1 - 279.T + 6.89e4T^{2} \)
43 \( 1 + (15.7 + 15.7i)T + 7.95e4iT^{2} \)
47 \( 1 + 470.T + 1.03e5T^{2} \)
53 \( 1 + (112. + 112. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-241. + 241. i)T - 2.05e5iT^{2} \)
61 \( 1 + (6.00 + 6.00i)T + 2.26e5iT^{2} \)
67 \( 1 + (273. - 273. i)T - 3.00e5iT^{2} \)
71 \( 1 + 448. iT - 3.57e5T^{2} \)
73 \( 1 - 54.7iT - 3.89e5T^{2} \)
79 \( 1 + 29.1iT - 4.93e5T^{2} \)
83 \( 1 + (893. + 893. i)T + 5.71e5iT^{2} \)
89 \( 1 - 281.T + 7.04e5T^{2} \)
97 \( 1 - 188.T + 9.12e5T^{2} \)
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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

−9.803263337176787899557161892380, −8.952220651615981652086935006309, −7.84097788829484475639149884872, −7.14013105580209431945229408067, −6.30324761608819598419768077839, −5.45876042942982685017685453298, −4.63085267521822879306671702289, −3.22677491597735791221714657306, −2.19527863617966045217731748133, −1.54242363112421402452087566426, 0.73576208920655169454154454570, 1.61032596731086374573032355722, 2.53603060714590362875301188858, 4.28517006221206071855790470361, 4.93603809957554652089463748252, 5.75148484964378276920418934165, 6.34359127219827821217194386868, 8.004858591823450810423420757225, 8.402199355427563100266738692489, 8.981221115452800961154238880333

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