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Properties

Label 2-1134-21.20-c1-0-27
Degree $2$
Conductor $1134$
Sign $0.923 + 0.383i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
Motivic weight $1$
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 − 4-s + 3.89·5-s + (−2.44 − 1.01i)7-s − i·8-s + 3.89i·10-s − 3.94i·11-s − 2.84i·13-s + (1.01 − 2.44i)14-s + 16-s − 0.742·17-s − 1.78i·19-s − 3.89·20-s + 3.94·22-s − 6.25i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.74·5-s + (−0.923 − 0.383i)7-s − 0.353i·8-s + 1.23i·10-s − 1.18i·11-s − 0.790i·13-s + (0.270 − 0.653i)14-s + 0.250·16-s − 0.179·17-s − 0.409i·19-s − 0.870·20-s + 0.840·22-s − 1.30i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.923 + 0.383i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.730508497\)
\(L(\frac12)\) \(\approx\) \(1.730508497\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (2.44 + 1.01i)T \)
good5 \( 1 - 3.89T + 5T^{2} \)
11 \( 1 + 3.94iT - 11T^{2} \)
13 \( 1 + 2.84iT - 13T^{2} \)
17 \( 1 + 0.742T + 17T^{2} \)
19 \( 1 + 1.78iT - 19T^{2} \)
23 \( 1 + 6.25iT - 23T^{2} \)
29 \( 1 - 2.88iT - 29T^{2} \)
31 \( 1 + 3.51iT - 31T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 0.943T + 43T^{2} \)
47 \( 1 - 2.18T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 0.0211T + 59T^{2} \)
61 \( 1 + 2.46iT - 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 - 4.85iT - 73T^{2} \)
79 \( 1 - 3.63T + 79T^{2} \)
83 \( 1 + 8.05T + 83T^{2} \)
89 \( 1 - 9.26T + 89T^{2} \)
97 \( 1 - 18.8iT - 97T^{2} \)
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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

−9.736513941657682797451419986136, −8.940375823643833302882631517424, −8.264429590755151765382396117026, −6.95156743874265647058037854219, −6.33282217066875981948107515308, −5.76103907568925774093683436640, −4.92494748808347970566702286465, −3.47425000623850896461080584741, −2.48832711061578435620464892783, −0.76050480176760372911013316719, 1.64931984764646837696577715002, 2.28777239198207835566728652003, 3.43992488823699468937333247908, 4.72828727573585912271098889623, 5.60309174626758655145135022249, 6.39812901131305518694814857678, 7.19563420260139764271790936844, 8.663715509993666112203014971325, 9.430668187955149703698413819429, 9.824332425495348262775802423166

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