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Properties

Label 2-1152-72.13-c1-0-9
Degree $2$
Conductor $1152$
Sign $0.996 + 0.0849i$
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  + (−1.37 + 1.05i)3-s + (−3.35 − 1.93i)5-s + (1.14 + 1.99i)7-s + (0.770 − 2.89i)9-s + (−3.60 + 2.08i)11-s + (0.846 + 0.488i)13-s + (6.65 − 0.884i)15-s − 6.79·17-s − 0.729i·19-s + (−3.68 − 1.52i)21-s + (2.94 − 5.10i)23-s + (5.02 + 8.69i)25-s + (2.00 + 4.79i)27-s + (4.32 − 2.49i)29-s + (1.53 − 2.65i)31-s + ⋯
L(s)  = 1  + (−0.792 + 0.609i)3-s + (−1.50 − 0.867i)5-s + (0.434 + 0.752i)7-s + (0.256 − 0.966i)9-s + (−1.08 + 0.627i)11-s + (0.234 + 0.135i)13-s + (1.71 − 0.228i)15-s − 1.64·17-s − 0.167i·19-s + (−0.803 − 0.331i)21-s + (0.614 − 1.06i)23-s + (1.00 + 1.73i)25-s + (0.385 + 0.922i)27-s + (0.803 − 0.463i)29-s + (0.275 − 0.476i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.996 + 0.0849i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.996 + 0.0849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6690130677\)
\(L(\frac12)\) \(\approx\) \(0.6690130677\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.37 - 1.05i)T \)
good5 \( 1 + (3.35 + 1.93i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.14 - 1.99i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.60 - 2.08i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.846 - 0.488i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.79T + 17T^{2} \)
19 \( 1 + 0.729iT - 19T^{2} \)
23 \( 1 + (-2.94 + 5.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.32 + 2.49i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.53 + 2.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.11iT - 37T^{2} \)
41 \( 1 + (3.12 - 5.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.80 + 2.77i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.86 - 8.42i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.81iT - 53T^{2} \)
59 \( 1 + (-10.6 - 6.14i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.8 + 6.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0524 + 0.0302i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + 1.71T + 73T^{2} \)
79 \( 1 + (5.77 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.35 - 3.66i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + (4.66 + 8.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.810930006658532398428428755464, −8.720312566623840088315875935940, −8.431354184516700445877767661184, −7.33750788061045629457805807418, −6.39877517021253013512607785234, −5.13839935554324976838754520296, −4.68911855647198498062564586163, −4.01693581656191732565127579837, −2.50544945001392143080264201916, −0.54684569434580012620972360427, 0.70983474642353202523839182509, 2.50949121422048293626089715639, 3.69520667519345847862008138897, 4.60434113151215888781126661427, 5.57915895042497202546796728362, 6.87352986513962220310417454013, 7.13535753855306319744520362583, 8.008703205118093987059716713567, 8.590058470422453640863036362772, 10.31758156095601932680896377037

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