↑

Properties

Label 2-552-69.68-c1-0-2
Degree $2$
Conductor $552$
Sign $-0.751 - 0.660i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 + 0.618i)3-s − 0.874·5-s + 1.41i·7-s + (2.23 − 2.00i)9-s + 5.11·11-s − 3.23·13-s + (1.41 − 0.540i)15-s − 4.91·17-s + 8.27i·19-s + (−0.874 − 2.28i)21-s + (−4.24 + 2.23i)23-s − 4.23·25-s + (−2.38 + 4.61i)27-s − 5.23i·29-s − 6·31-s + ⋯
L(s)  = 1  + (−0.934 + 0.356i)3-s − 0.390·5-s + 0.534i·7-s + (0.745 − 0.666i)9-s + 1.54·11-s − 0.897·13-s + (0.365 − 0.139i)15-s − 1.19·17-s + 1.89i·19-s + (−0.190 − 0.499i)21-s + (−0.884 + 0.466i)23-s − 0.847·25-s + (−0.458 + 0.888i)27-s − 0.972i·29-s − 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.751 - 0.660i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.751 - 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197563 + 0.524169i\)
\(L(\frac12)\) \(\approx\) \(0.197563 + 0.524169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.61 - 0.618i)T \)
23 \( 1 + (4.24 - 2.23i)T \)
good5 \( 1 + 0.874T + 5T^{2} \)
7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 - 5.11T + 11T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 + 4.91T + 17T^{2} \)
19 \( 1 - 8.27iT - 19T^{2} \)
29 \( 1 + 5.23iT - 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 10.7iT - 37T^{2} \)
41 \( 1 - 0.472iT - 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 - 0.874T + 53T^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 + 1.20iT - 61T^{2} \)
67 \( 1 + 6.53iT - 67T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + 7.23T + 73T^{2} \)
79 \( 1 + 6.65iT - 79T^{2} \)
83 \( 1 + 7.94T + 83T^{2} \)
89 \( 1 - 0.333T + 89T^{2} \)
97 \( 1 + 2.41iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.44858685851881199808720967857, −10.10965937362194059206645575707, −9.620338448648443669825175153478, −8.556253077339284727752936124116, −7.42475414452967052396264675274, −6.37601774940551165236674752199, −5.73191377437448414482299424190, −4.43329276268403949784208836061, −3.74188980756035717074216153232, −1.76448485705016061065564578608, 0.35858432026172426610123530756, 2.05940171217713392051953755433, 3.98369827803966235198981914841, 4.68119015946928778833445014550, 5.94078079314602353153722979758, 7.05642359596689663178653356342, 7.23436270532500575385724476566, 8.792624759657184618209407196659, 9.559484211794862741469528414610, 10.79653691592681419685147513527

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