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Properties

Label 2-1152-96.59-c1-0-1
Degree $2$
Conductor $1152$
Sign $-0.977 + 0.212i$
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  + (−1.39 + 3.36i)5-s + (1.05 − 1.05i)7-s + (−1.50 + 3.64i)11-s + (−2.23 + 0.927i)13-s − 7.49·17-s + (−0.818 − 1.97i)19-s + (5.80 − 5.80i)23-s + (−5.86 − 5.86i)25-s + (0.326 − 0.135i)29-s − 1.71i·31-s + (2.08 + 5.03i)35-s + (0.387 + 0.160i)37-s + (−1.50 − 1.50i)41-s + (−7.40 − 3.06i)43-s + 7.27i·47-s + ⋯
L(s)  = 1  + (−0.624 + 1.50i)5-s + (0.399 − 0.399i)7-s + (−0.455 + 1.09i)11-s + (−0.620 + 0.257i)13-s − 1.81·17-s + (−0.187 − 0.453i)19-s + (1.21 − 1.21i)23-s + (−1.17 − 1.17i)25-s + (0.0606 − 0.0251i)29-s − 0.307i·31-s + (0.352 + 0.851i)35-s + (0.0636 + 0.0263i)37-s + (−0.234 − 0.234i)41-s + (−1.12 − 0.467i)43-s + 1.06i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.977 + 0.212i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.977 + 0.212i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3778430923\)
\(L(\frac12)\) \(\approx\) \(0.3778430923\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.39 - 3.36i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.05 + 1.05i)T - 7iT^{2} \)
11 \( 1 + (1.50 - 3.64i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.23 - 0.927i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 7.49T + 17T^{2} \)
19 \( 1 + (0.818 + 1.97i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.80 + 5.80i)T - 23iT^{2} \)
29 \( 1 + (-0.326 + 0.135i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 1.71iT - 31T^{2} \)
37 \( 1 + (-0.387 - 0.160i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.50 + 1.50i)T + 41iT^{2} \)
43 \( 1 + (7.40 + 3.06i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 7.27iT - 47T^{2} \)
53 \( 1 + (-3.94 - 1.63i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (12.7 + 5.30i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (0.579 + 1.40i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-7.96 + 3.29i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (4.75 + 4.75i)T + 71iT^{2} \)
73 \( 1 + (7.99 - 7.99i)T - 73iT^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 + (1.35 - 0.560i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (4.75 - 4.75i)T - 89iT^{2} \)
97 \( 1 + 1.28T + 97T^{2} \)
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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

−10.49757836436380690085253867850, −9.517157521206224765046838267194, −8.520026831673306182906511857040, −7.48539258932743092177014889554, −7.00292072766388855402732372193, −6.43504451273530880810374972528, −4.77708198410856324101960993323, −4.30723305887718099379883403309, −2.92337127132541470413576557485, −2.18165820652745466001246251408, 0.15733103712833601722993394022, 1.60084239634475251926962014518, 3.06183369680371733340473241860, 4.29032090208021960974043779294, 5.04429695564716460623626733338, 5.67849517267352497400048105962, 6.97783437303562615965858956660, 7.959756741697969920608613319179, 8.690551481139663667182176946052, 8.951238354868253897389594457877

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