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Properties

Label 2-1161-9.4-c1-0-41
Degree $2$
Conductor $1161$
Sign $-0.217 - 0.975i$
Analytic cond. $9.27063$
Root an. cond. $3.04477$
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  + (−0.887 − 1.53i)2-s + (−0.575 + 0.996i)4-s + (1.84 − 3.20i)5-s + (0.0448 + 0.0776i)7-s − 1.50·8-s − 6.56·10-s + (−2.40 − 4.16i)11-s + (−2.20 + 3.81i)13-s + (0.0796 − 0.137i)14-s + (2.48 + 4.31i)16-s − 4.51·17-s + 0.457·19-s + (2.12 + 3.68i)20-s + (−4.26 + 7.39i)22-s + (−3.13 + 5.43i)23-s + ⋯
L(s)  = 1  + (−0.627 − 1.08i)2-s + (−0.287 + 0.498i)4-s + (0.827 − 1.43i)5-s + (0.0169 + 0.0293i)7-s − 0.532·8-s − 2.07·10-s + (−0.725 − 1.25i)11-s + (−0.610 + 1.05i)13-s + (0.0212 − 0.0368i)14-s + (0.622 + 1.07i)16-s − 1.09·17-s + 0.104·19-s + (0.475 + 0.824i)20-s + (−0.910 + 1.57i)22-s + (−0.654 + 1.13i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1161\)    =    \(3^{3} \cdot 43\)
Sign: $-0.217 - 0.975i$
Analytic conductor: \(9.27063\)
Root analytic conductor: \(3.04477\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1161} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1161,\ (\ :1/2),\ -0.217 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5211615618\)
\(L(\frac12)\) \(\approx\) \(0.5211615618\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.887 + 1.53i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.84 + 3.20i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.0448 - 0.0776i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.40 + 4.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.20 - 3.81i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.51T + 17T^{2} \)
19 \( 1 - 0.457T + 19T^{2} \)
23 \( 1 + (3.13 - 5.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.41 - 4.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.62 + 2.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.48T + 37T^{2} \)
41 \( 1 + (-3.54 + 6.13i)T + (-20.5 - 35.5i)T^{2} \)
47 \( 1 + (-1.98 - 3.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.18T + 53T^{2} \)
59 \( 1 + (-4.89 + 8.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.86 + 10.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.11 + 5.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + (-4.64 - 8.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.01 - 3.49i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.46T + 89T^{2} \)
97 \( 1 + (-0.339 - 0.588i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.217174859195628862719566098104, −8.815379184942166629693017167052, −8.033563886316847313293266922835, −6.57551281763778028942629659714, −5.63641046292286743977164762277, −4.92295409132553889903793206681, −3.66951583558792699685630379237, −2.33672330381172544198535107868, −1.57802034608532165551994327660, −0.25687979280010517274624877298, 2.32329510740861930742845797885, 2.91492841254996246408199455268, 4.61764177651993399822280488812, 5.70332425807347898244382931798, 6.44156472843129880047627172374, 7.10713661650544211052436514127, 7.66275236217955676857186495274, 8.575269544254339250454843608007, 9.611580504338224594485288296559, 10.25984101306485049059923029840

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