Properties

Label 2-552-184.85-c1-0-20
Degree $2$
Conductor $552$
Sign $-0.424 - 0.905i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0519 + 1.41i)2-s + (0.540 + 0.841i)3-s + (−1.99 − 0.146i)4-s + (2.13 − 0.975i)5-s + (−1.21 + 0.720i)6-s + (0.595 − 0.174i)7-s + (0.311 − 2.81i)8-s + (−0.415 + 0.909i)9-s + (1.26 + 3.06i)10-s + (−1.11 + 0.963i)11-s + (−0.954 − 1.75i)12-s + (−0.995 + 3.38i)13-s + (0.216 + 0.850i)14-s + (1.97 + 1.26i)15-s + (3.95 + 0.586i)16-s + (−1.11 + 7.76i)17-s + ⋯
L(s)  = 1  + (−0.0367 + 0.999i)2-s + (0.312 + 0.485i)3-s + (−0.997 − 0.0734i)4-s + (0.954 − 0.436i)5-s + (−0.496 + 0.294i)6-s + (0.225 − 0.0661i)7-s + (0.110 − 0.993i)8-s + (−0.138 + 0.303i)9-s + (0.400 + 0.970i)10-s + (−0.335 + 0.290i)11-s + (−0.275 − 0.507i)12-s + (−0.276 + 0.940i)13-s + (0.0577 + 0.227i)14-s + (0.509 + 0.327i)15-s + (0.989 + 0.146i)16-s + (−0.270 + 1.88i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.424 - 0.905i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.424 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.883014 + 1.38974i\)
\(L(\frac12)\) \(\approx\) \(0.883014 + 1.38974i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0519 - 1.41i)T \)
3 \( 1 + (-0.540 - 0.841i)T \)
23 \( 1 + (-4.79 - 0.0498i)T \)
good5 \( 1 + (-2.13 + 0.975i)T + (3.27 - 3.77i)T^{2} \)
7 \( 1 + (-0.595 + 0.174i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (1.11 - 0.963i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (0.995 - 3.38i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (1.11 - 7.76i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-6.09 + 0.876i)T + (18.2 - 5.35i)T^{2} \)
29 \( 1 + (-1.45 - 0.209i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (-5.62 - 3.61i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (9.90 + 4.52i)T + (24.2 + 27.9i)T^{2} \)
41 \( 1 + (1.33 + 2.92i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (5.83 + 9.08i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + (-0.599 - 2.04i)T + (-44.5 + 28.6i)T^{2} \)
59 \( 1 + (-2.33 + 7.95i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-6.43 + 10.0i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (8.28 + 7.18i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (5.25 - 6.06i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (1.40 + 9.73i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-4.20 - 1.23i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-1.42 - 0.652i)T + (54.3 + 62.7i)T^{2} \)
89 \( 1 + (-5.06 + 3.25i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (4.42 + 9.67i)T + (-63.5 + 73.3i)T^{2} \)
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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

−10.64815268994646091776642717979, −9.972641287559965713648117159412, −9.089505627912562101091343457098, −8.605851868100061573780277853439, −7.45881395425708443386668889492, −6.51607879644329018912627757558, −5.42312110746585027248614030585, −4.80636674298805053271244841785, −3.60251056191305100929354553595, −1.72549013644998965484222490300, 1.04988290985780594469061072046, 2.61086463287442861926739977228, 3.06152414044954382150877461224, 4.90573696310129620843470239664, 5.62866997583383092804493932062, 7.01417128830035942607445053232, 7.965073005181159536652567066771, 8.967306241114536920107324434894, 9.822986238016046560645750862499, 10.34909539878275579475390909198

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