Properties

Label 2-1161-129.122-c0-0-3
Degree $2$
Conductor $1161$
Sign $0.300 - 0.953i$
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  + i·2-s + (1.40 − 0.809i)5-s + (−0.309 + 0.535i)7-s + i·8-s + (0.809 + 1.40i)10-s + 1.61i·11-s + (0.809 − 1.40i)13-s + (−0.535 − 0.309i)14-s − 16-s + (−1.40 − 0.809i)17-s + (−0.309 − 0.535i)19-s − 1.61·22-s + (−0.535 + 0.309i)23-s + (0.809 − 1.40i)25-s + (1.40 + 0.809i)26-s + ⋯
L(s)  = 1  + i·2-s + (1.40 − 0.809i)5-s + (−0.309 + 0.535i)7-s + i·8-s + (0.809 + 1.40i)10-s + 1.61i·11-s + (0.809 − 1.40i)13-s + (−0.535 − 0.309i)14-s − 16-s + (−1.40 − 0.809i)17-s + (−0.309 − 0.535i)19-s − 1.61·22-s + (−0.535 + 0.309i)23-s + (0.809 − 1.40i)25-s + (1.40 + 0.809i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1161\)    =    \(3^{3} \cdot 43\)
Sign: $0.300 - 0.953i$
Analytic conductor: \(0.579414\)
Root analytic conductor: \(0.761192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1161} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1161,\ (\ :0),\ 0.300 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.393291887\)
\(L(\frac12)\) \(\approx\) \(1.393291887\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 - iT - T^{2} \)
5 \( 1 + (-1.40 + 0.809i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 - 1.61iT - T^{2} \)
13 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (1.40 + 0.809i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.535 - 0.309i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - 0.618iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.535 + 0.309i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + 0.618iT - T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.61T + T^{2} \)
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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

−9.935436817593949735026410755179, −9.135074913497067855148915191731, −8.604891829641042658644181894917, −7.56504646294156340155831746979, −6.67340880844829369424585729968, −5.99709904102110869506606644171, −5.27394560238537368144732068479, −4.59739993480483687202029735520, −2.64725495857878167579861452491, −1.85408965094289176994301350777, 1.54107757652419307287380007244, 2.36441836502158253006420749801, 3.42976903320511324576292803656, 4.19540062594556646775632826386, 6.04735708983201414059442426057, 6.26970353906060181851784600810, 7.01627413301465205729240159166, 8.566613953841310767559214957257, 9.197868698032742772079902862308, 10.15004872775300703780860495784

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