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Properties

Label 2-1148-1148.163-c0-0-5
Degree $2$
Conductor $1148$
Sign $0.0633 + 0.997i$
Analytic cond. $0.572926$
Root an. cond. $0.756919$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (0.499 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s + (0.499 + 0.866i)14-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)19-s − 0.999·20-s − 0.999·21-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (0.499 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s + (0.499 + 0.866i)14-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)19-s − 0.999·20-s − 0.999·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(0.572926\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :0),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6984517872\)
\(L(\frac12)\) \(\approx\) \(0.6984517872\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 - T \)
good3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.746447742330723522468512071800, −8.662417026945543986841090108095, −8.160775831827835085407142989910, −7.29045281181377241814306036779, −6.54556642574320883872358064586, −5.70085743290770348017164171451, −5.06437340421326642834258678823, −3.91173057400785838861138862363, −1.73217175561221329925690682096, −0.827225766858395063243323841899, 2.07001008651529878065695396188, 2.69498805889845761080913673024, 4.11489247448533858616863579555, 4.89034434730231048082670550569, 5.75550773140910433551476629299, 7.03541724368761814260918152495, 7.85290978032906959231405822599, 8.969755346835513017661856541051, 9.538139982747020315127533888076, 10.37321964676910319257055147918

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