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Properties

Label 2-1161-129.92-c0-0-0
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: $\chi_{1161} (350, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1161,\ (\ :0),\ -0.300 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8925206424\)
\(L(\frac12)\) \(\approx\) \(0.8925206424\)
\(L(1)\) 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

−10.03730006982485508718568831069, −9.198064775515209245618075122849, −8.321667493052630289703597428314, −7.54894818582857522370643945571, −7.16857259399505643543453984167, −6.28481218198297676741949288101, −4.98345552271457445147318212756, −4.43506787798890369932619542900, −3.42358721837135416809936593045, −1.60500476550645853405808498542, 0.867320819377428045466947014368, 2.85301223097835147413910501067, 3.27839130984698103478532574337, 3.90839399965450219923337504182, 5.61453153308669477894110738204, 6.34259255835391648635500364025, 7.39929195337402563549030714316, 8.197259902755746645057898975641, 8.847980017336487282567737374678, 10.21226629820629425640855755163

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