Properties

Label 2-1148-1148.655-c0-0-6
Degree $2$
Conductor $1148$
Sign $0.660 + 0.750i$
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 + (0.258 − 0.448i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 1.5i)5-s + 0.517·6-s + (−0.965 − 0.258i)7-s − 0.999·8-s + (0.366 + 0.633i)9-s + (0.866 − 1.5i)10-s + (0.965 − 1.67i)11-s + (0.258 + 0.448i)12-s + (−0.258 − 0.965i)14-s − 0.896·15-s + (−0.5 − 0.866i)16-s + (−0.366 + 0.633i)18-s + (−0.965 − 1.67i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.258 − 0.448i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 1.5i)5-s + 0.517·6-s + (−0.965 − 0.258i)7-s − 0.999·8-s + (0.366 + 0.633i)9-s + (0.866 − 1.5i)10-s + (0.965 − 1.67i)11-s + (0.258 + 0.448i)12-s + (−0.258 − 0.965i)14-s − 0.896·15-s + (−0.5 − 0.866i)16-s + (−0.366 + 0.633i)18-s + (−0.965 − 1.67i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.660 + 0.750i$
Analytic conductor: \(0.572926\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :0),\ 0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001140863\)
\(L(\frac12)\) \(\approx\) \(1.001140863\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 - T \)
good3 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 0.517T + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.281715104094288726791718873663, −8.868998233162937466676745851911, −8.238615381514168088135348767588, −7.41363987752470024825899005507, −6.58157952463568203381390544560, −5.73421378255363673077359962603, −4.60471448050339605190701809929, −4.03576882114875450705771651254, −2.94684129174437532275751933013, −0.76277230857141769827968820270, 2.03759153083777961913956844131, 3.15870938995719671389297045974, 3.89126020310432153426250619893, 4.31470339002634743473703539665, 6.06081160318513268323585271249, 6.64380353647122783945187504257, 7.43007179956295719388764162838, 8.828707158511858777392123242960, 9.675792138844511465544592462585, 10.17268815689527154751327040817

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