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Properties

Label 2-1183-91.86-c0-0-0
Degree $2$
Conductor $1183$
Sign $-0.580 - 0.814i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
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  + (0.965 − 0.258i)2-s + (−0.5 − 0.866i)3-s + (−0.965 + 0.258i)5-s + (−0.707 − 0.707i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)10-s + (−0.965 − 0.258i)11-s + (−0.500 + 0.866i)14-s + (0.707 + 0.707i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.965 + 0.258i)19-s + (0.965 + 0.258i)21-s − 22-s + (0.866 + 0.5i)23-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.5 − 0.866i)3-s + (−0.965 + 0.258i)5-s + (−0.707 − 0.707i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)10-s + (−0.965 − 0.258i)11-s + (−0.500 + 0.866i)14-s + (0.707 + 0.707i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.965 + 0.258i)19-s + (0.965 + 0.258i)21-s − 22-s + (0.866 + 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.580 - 0.814i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ -0.580 - 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1863788100\)
\(L(\frac12)\) \(\approx\) \(0.1863788100\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 \)
good2 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
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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

−10.57534138549700273824701254905, −9.343521569521577140530785161779, −8.437954089328496844360530336402, −7.74362301488384298923381877507, −6.65367754488528289439482793107, −6.07238930210090469809128350241, −5.16864395241114334128894798503, −4.09185815997014302375133492562, −3.25472734284793241518151277164, −2.24510670294923050317986348610, 0.12023736662908658976329524380, 2.88285712957823136508452782717, 3.95161560093229464119157194336, 4.56317365350444066851361557689, 5.00543379792759014038151019018, 6.16948503291127696767014115328, 6.99311931819981400882675664581, 7.87959902345260126107891311017, 9.041941323856647032222566249603, 9.735951490967885051771759096840

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