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Properties

Label 2-1152-48.5-c2-0-12
Degree $2$
Conductor $1152$
Sign $0.578 - 0.815i$
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  + (−1.92 + 1.92i)5-s + 2.43i·7-s + (−2.79 + 2.79i)11-s + (1.54 − 1.54i)13-s − 20.3i·17-s + (25.0 − 25.0i)19-s + 3.55·23-s + 17.5i·25-s + (23.4 + 23.4i)29-s − 13.3·31-s + (−4.68 − 4.68i)35-s + (25.4 + 25.4i)37-s − 64.0·41-s + (24.6 + 24.6i)43-s + 79.5i·47-s + ⋯
L(s)  = 1  + (−0.385 + 0.385i)5-s + 0.347i·7-s + (−0.253 + 0.253i)11-s + (0.118 − 0.118i)13-s − 1.19i·17-s + (1.31 − 1.31i)19-s + 0.154·23-s + 0.702i·25-s + (0.808 + 0.808i)29-s − 0.429·31-s + (−0.133 − 0.133i)35-s + (0.687 + 0.687i)37-s − 1.56·41-s + (0.573 + 0.573i)43-s + 1.69i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.578 - 0.815i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.637959700\)
\(L(\frac12)\) \(\approx\) \(1.637959700\)
\(L(2)\) 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 \( 1 + (1.92 - 1.92i)T - 25iT^{2} \)
7 \( 1 - 2.43iT - 49T^{2} \)
11 \( 1 + (2.79 - 2.79i)T - 121iT^{2} \)
13 \( 1 + (-1.54 + 1.54i)T - 169iT^{2} \)
17 \( 1 + 20.3iT - 289T^{2} \)
19 \( 1 + (-25.0 + 25.0i)T - 361iT^{2} \)
23 \( 1 - 3.55T + 529T^{2} \)
29 \( 1 + (-23.4 - 23.4i)T + 841iT^{2} \)
31 \( 1 + 13.3T + 961T^{2} \)
37 \( 1 + (-25.4 - 25.4i)T + 1.36e3iT^{2} \)
41 \( 1 + 64.0T + 1.68e3T^{2} \)
43 \( 1 + (-24.6 - 24.6i)T + 1.84e3iT^{2} \)
47 \( 1 - 79.5iT - 2.20e3T^{2} \)
53 \( 1 + (39.8 - 39.8i)T - 2.80e3iT^{2} \)
59 \( 1 + (-19.2 + 19.2i)T - 3.48e3iT^{2} \)
61 \( 1 + (63.5 - 63.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (-65.1 + 65.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 84.6T + 5.04e3T^{2} \)
73 \( 1 - 39.7iT - 5.32e3T^{2} \)
79 \( 1 - 109.T + 6.24e3T^{2} \)
83 \( 1 + (-14.7 - 14.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 32.3T + 7.92e3T^{2} \)
97 \( 1 - 123.T + 9.40e3T^{2} \)
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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

−9.512861752884463105931929631419, −9.133136502607586675173447280481, −7.936572850136828662632222513656, −7.27940967348517664533373924970, −6.54962728140755621242551673783, −5.29942228020773860224218812952, −4.72731962712442987046762862833, −3.28934035733566657180103303080, −2.66104832842868325302084682808, −1.00012288400851266074415325814, 0.62106719101839372825672933021, 1.92184106358123393134016730966, 3.42867909797153177275904345706, 4.10620280045561649439752557840, 5.23810856066777108084504878639, 6.05184093634052257546695304057, 7.05061400807357290838077300351, 8.070713433853700603466413753832, 8.385528372132821039246462716199, 9.578317370580568556650647491351

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