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Properties

Label 2-1152-48.11-c1-0-0
Degree $2$
Conductor $1152$
Sign $-0.999 - 0.0355i$
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  + (−2.63 + 2.63i)5-s − 0.207·7-s + (3.66 + 3.66i)11-s + (−0.255 + 0.255i)13-s + 0.654i·17-s + (−4.46 − 4.46i)19-s + 3.48i·23-s − 8.86i·25-s + (−4.33 − 4.33i)29-s + 6.16i·31-s + (0.545 − 0.545i)35-s + (−4.39 − 4.39i)37-s + 0.0684·41-s + (−5.65 + 5.65i)43-s − 9.14·47-s + ⋯
L(s)  = 1  + (−1.17 + 1.17i)5-s − 0.0783·7-s + (1.10 + 1.10i)11-s + (−0.0708 + 0.0708i)13-s + 0.158i·17-s + (−1.02 − 1.02i)19-s + 0.727i·23-s − 1.77i·25-s + (−0.805 − 0.805i)29-s + 1.10i·31-s + (0.0921 − 0.0921i)35-s + (−0.722 − 0.722i)37-s + 0.0106·41-s + (−0.862 + 0.862i)43-s − 1.33·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.999 - 0.0355i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.999 - 0.0355i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5172348692\)
\(L(\frac12)\) \(\approx\) \(0.5172348692\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 2.63i)T - 5iT^{2} \)
7 \( 1 + 0.207T + 7T^{2} \)
11 \( 1 + (-3.66 - 3.66i)T + 11iT^{2} \)
13 \( 1 + (0.255 - 0.255i)T - 13iT^{2} \)
17 \( 1 - 0.654iT - 17T^{2} \)
19 \( 1 + (4.46 + 4.46i)T + 19iT^{2} \)
23 \( 1 - 3.48iT - 23T^{2} \)
29 \( 1 + (4.33 + 4.33i)T + 29iT^{2} \)
31 \( 1 - 6.16iT - 31T^{2} \)
37 \( 1 + (4.39 + 4.39i)T + 37iT^{2} \)
41 \( 1 - 0.0684T + 41T^{2} \)
43 \( 1 + (5.65 - 5.65i)T - 43iT^{2} \)
47 \( 1 + 9.14T + 47T^{2} \)
53 \( 1 + (1.51 - 1.51i)T - 53iT^{2} \)
59 \( 1 + (2.53 + 2.53i)T + 59iT^{2} \)
61 \( 1 + (-5.46 + 5.46i)T - 61iT^{2} \)
67 \( 1 + (-4.77 - 4.77i)T + 67iT^{2} \)
71 \( 1 - 5.94iT - 71T^{2} \)
73 \( 1 + 6.93iT - 73T^{2} \)
79 \( 1 + 4.72iT - 79T^{2} \)
83 \( 1 + (4.32 - 4.32i)T - 83iT^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 0.925T + 97T^{2} \)
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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

−10.21455168852978424387070590113, −9.463776547668230579759398152058, −8.488319514572798971501251128876, −7.56606635377392050024601862027, −6.88602485038162189957704139616, −6.40714716793297599025510588385, −4.84087984911850865139689276472, −3.99254491215169947271854755851, −3.21988983279751241620151267583, −1.92122866457388271194807715436, 0.22777977939297146878502828449, 1.55011532901290226573344933329, 3.42300318365579971140135300536, 4.04025203369675209526616133245, 4.95989085525439557288819122645, 6.00134072823349161855192725847, 6.90728920365888292478753643628, 8.080330488199918488552892787451, 8.473825716993579267263376889234, 9.140501203960249981576626471651

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