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Properties

Label 2-1152-16.11-c2-0-22
Degree $2$
Conductor $1152$
Sign $0.0366 + 0.999i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.24 − 5.24i)5-s + 5.32·7-s + (−12.2 + 12.2i)11-s + (5.73 − 5.73i)13-s + 23.3·17-s + (11.7 + 11.7i)19-s + 5.80·23-s + 29.9i·25-s + (18.3 − 18.3i)29-s − 16.9i·31-s + (−27.9 − 27.9i)35-s + (−15.3 − 15.3i)37-s − 29.2i·41-s + (33.4 − 33.4i)43-s + 18.2i·47-s + ⋯
L(s)  = 1  + (−1.04 − 1.04i)5-s + 0.761·7-s + (−1.11 + 1.11i)11-s + (0.441 − 0.441i)13-s + 1.37·17-s + (0.618 + 0.618i)19-s + 0.252·23-s + 1.19i·25-s + (0.634 − 0.634i)29-s − 0.545i·31-s + (−0.798 − 0.798i)35-s + (−0.414 − 0.414i)37-s − 0.713i·41-s + (0.776 − 0.776i)43-s + 0.387i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0366 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0366 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.0366 + 0.999i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.0366 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.408829372\)
\(L(\frac12)\) \(\approx\) \(1.408829372\)
\(L(2)\) 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 \( 1 + (5.24 + 5.24i)T + 25iT^{2} \)
7 \( 1 - 5.32T + 49T^{2} \)
11 \( 1 + (12.2 - 12.2i)T - 121iT^{2} \)
13 \( 1 + (-5.73 + 5.73i)T - 169iT^{2} \)
17 \( 1 - 23.3T + 289T^{2} \)
19 \( 1 + (-11.7 - 11.7i)T + 361iT^{2} \)
23 \( 1 - 5.80T + 529T^{2} \)
29 \( 1 + (-18.3 + 18.3i)T - 841iT^{2} \)
31 \( 1 + 16.9iT - 961T^{2} \)
37 \( 1 + (15.3 + 15.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 29.2iT - 1.68e3T^{2} \)
43 \( 1 + (-33.4 + 33.4i)T - 1.84e3iT^{2} \)
47 \( 1 - 18.2iT - 2.20e3T^{2} \)
53 \( 1 + (66.9 + 66.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-27.1 + 27.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (65.2 - 65.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (37.6 + 37.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 42.6T + 5.04e3T^{2} \)
73 \( 1 + 106. iT - 5.32e3T^{2} \)
79 \( 1 + 21.2iT - 6.24e3T^{2} \)
83 \( 1 + (24.1 + 24.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 52.8iT - 7.92e3T^{2} \)
97 \( 1 + 21.0T + 9.40e3T^{2} \)
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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

−9.389044416154819182012885671600, −8.258491416311611873826613448296, −7.903604258913437294028794776069, −7.32896832252059674616280447760, −5.71796441287114661479290108723, −5.03134454426264212919540849260, −4.31296981006184400656638967746, −3.24241557567408830919388704352, −1.72735917975428119197558900887, −0.51161661053287010603125225613, 1.09516944421783364016493597717, 2.91032796154575853617951085538, 3.34371310879663669293176914825, 4.62864879551224039214491726232, 5.53143237284752978476030351343, 6.57604846832952232869435377457, 7.54586775839426601179246885452, 7.978827266987797007789666838646, 8.764576152216458684081825981479, 9.997086511212282115963524769324

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