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Properties

Label 2-1161-129.92-c0-0-6
Degree $2$
Conductor $1161$
Sign $-0.736 - 0.675i$
Analytic cond. $0.579414$
Root an. cond. $0.761192$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 1.00·4-s + (−1.22 − 0.707i)5-s + (−1.00 + 1.73i)10-s + (−0.5 − 0.866i)13-s − 0.999·16-s + (−0.5 + 0.866i)19-s + (1.22 + 0.707i)20-s + (−1.22 − 0.707i)23-s + (0.499 + 0.866i)25-s + (−1.22 + 0.707i)26-s + (0.5 − 0.866i)31-s + 1.41i·32-s + (1.22 + 0.707i)38-s + 1.41i·41-s + ⋯
L(s)  = 1  − 1.41i·2-s − 1.00·4-s + (−1.22 − 0.707i)5-s + (−1.00 + 1.73i)10-s + (−0.5 − 0.866i)13-s − 0.999·16-s + (−0.5 + 0.866i)19-s + (1.22 + 0.707i)20-s + (−1.22 − 0.707i)23-s + (0.499 + 0.866i)25-s + (−1.22 + 0.707i)26-s + (0.5 − 0.866i)31-s + 1.41i·32-s + (1.22 + 0.707i)38-s + 1.41i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1161\)    =    \(3^{3} \cdot 43\)
Sign: $-0.736 - 0.675i$
Analytic conductor: \(0.579414\)
Root analytic conductor: \(0.761192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1161} (350, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1161,\ (\ :0),\ -0.736 - 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5566877333\)
\(L(\frac12)\) \(\approx\) \(0.5566877333\)
\(L(1)\) 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 \( 1 + T \)
good2 \( 1 + 1.41iT - T^{2} \)
5 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.914475256695483484529770435734, −8.618689731187393589407890797835, −8.174624910268619765341225336652, −7.22168374052198102475363896199, −5.94510129810113680670438137890, −4.67013023140588644540615574528, −4.05921518182423849996271495602, −3.18751999447018446239533973391, −1.99421473780104230310413314575, −0.47920580546862995674610945869, 2.41153202429023881317329278345, 3.79739479701340031840157837772, 4.57794938370808403510050646255, 5.60075474063147461218062270435, 6.73609168261057793332505248781, 7.02465490384362639219897797929, 7.85884375074586141983902114216, 8.495048047456296548539857652295, 9.381737336932254299567139537713, 10.46577853657183785785260008239

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