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Properties

Label 2-1183-91.11-c0-0-0
Degree $2$
Conductor $1183$
Sign $-0.641 - 0.767i$
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 + 3-s + (−0.258 + 0.965i)5-s + (−0.965 + 0.258i)6-s + (0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s − i·10-s + (−0.707 + 0.707i)11-s + (−0.499 − 0.866i)14-s + (−0.258 + 0.965i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.707 − 0.707i)19-s + (0.258 + 0.965i)21-s + (0.500 − 0.866i)22-s + (0.866 + 0.5i)23-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + 3-s + (−0.258 + 0.965i)5-s + (−0.965 + 0.258i)6-s + (0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s − i·10-s + (−0.707 + 0.707i)11-s + (−0.499 − 0.866i)14-s + (−0.258 + 0.965i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.707 − 0.707i)19-s + (0.258 + 0.965i)21-s + (0.500 − 0.866i)22-s + (0.866 + 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.641 - 0.767i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ -0.641 - 0.767i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6462262299\)
\(L(\frac12)\) \(\approx\) \(0.6462262299\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 \)
good2 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
3 \( 1 - T + T^{2} \)
5 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 - T + T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (-1.93 + 0.517i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.990206526726046168684085885925, −9.188699629934379196490195219291, −8.626952586446063624982190285297, −8.030964729299137382943634857594, −7.23909271722397167556006134281, −6.52034730261819024648668430783, −5.13611784127847764974333992168, −4.00746021266438724729793804716, −2.84285487084676817991284901099, −2.12264451276257881912407701599, 0.69298927756968005901700468680, 2.03924699872086419519639677515, 3.27831131597596203387959979300, 4.50853298557989856467512956198, 5.09054723624070673188185322906, 6.61361032662995716947192745107, 7.80838880055230692192450504574, 8.313568140299547101059577602495, 8.664223077751287865895603142749, 9.488882301084992331497989173283

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