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Properties

Label 2-1161-129.92-c0-0-2
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 + (0.535 + 0.309i)5-s + (0.809 + 1.40i)7-s + i·8-s + (−0.309 + 0.535i)10-s − 0.618i·11-s + (−0.309 − 0.535i)13-s + (−1.40 + 0.809i)14-s − 16-s + (−0.535 + 0.309i)17-s + (0.809 − 1.40i)19-s + 0.618·22-s + (−1.40 − 0.809i)23-s + (−0.309 − 0.535i)25-s + (0.535 − 0.309i)26-s + ⋯
L(s)  = 1  + i·2-s + (0.535 + 0.309i)5-s + (0.809 + 1.40i)7-s + i·8-s + (−0.309 + 0.535i)10-s − 0.618i·11-s + (−0.309 − 0.535i)13-s + (−1.40 + 0.809i)14-s − 16-s + (−0.535 + 0.309i)17-s + (0.809 − 1.40i)19-s + 0.618·22-s + (−1.40 − 0.809i)23-s + (−0.309 − 0.535i)25-s + (0.535 − 0.309i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.331241817\)
\(L(\frac12)\) \(\approx\) \(1.331241817\)
\(L(1)\) 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 \( 1 + (-0.535 - 0.309i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + 0.618iT - T^{2} \)
13 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.535 - 0.309i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.40 + 0.809i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.61iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1.40 - 0.809i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 - 1.61iT - T^{2} \)
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 \( 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + 0.618T + T^{2} \)
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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

−10.20630861160793010334705133632, −9.025826589475942406781519854705, −8.525390669854641762655398807733, −7.78038482994683114253657554239, −6.82335290377932177831457788952, −5.90584892571361168437670063141, −5.55619869823116604818805016192, −4.56891060241296614409543295324, −2.74083826748568944068672135663, −2.14002882726016571885559672979, 1.41155860770899583969212521682, 2.01174394344928704448951503952, 3.58786312166263280871750927507, 4.25020745765000525940001492280, 5.27079735955058880956072836470, 6.49946761533100789553331648770, 7.37510001251980287147023519306, 7.978038375883224155212405493478, 9.381254111946807268662702542557, 9.968612722283327105621416036071

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