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Properties

Label 2-990-55.9-c1-0-17
Degree $2$
Conductor $990$
Sign $0.887 + 0.460i$
Analytic cond. $7.90518$
Root an. cond. $2.81161$
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  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (2.16 − 0.546i)5-s + (−1.82 + 2.51i)7-s + (−0.587 − 0.809i)8-s + (−2.23 − 0.150i)10-s + (−0.565 − 3.26i)11-s + (0.595 + 0.193i)13-s + (2.51 − 1.82i)14-s + (0.309 + 0.951i)16-s + (−2.33 + 0.759i)17-s + (6.82 − 4.96i)19-s + (2.07 + 0.832i)20-s + (−0.471 + 3.28i)22-s − 2.36i·23-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.404 + 0.293i)4-s + (0.969 − 0.244i)5-s + (−0.689 + 0.948i)7-s + (−0.207 − 0.286i)8-s + (−0.705 − 0.0474i)10-s + (−0.170 − 0.985i)11-s + (0.165 + 0.0536i)13-s + (0.670 − 0.487i)14-s + (0.0772 + 0.237i)16-s + (−0.566 + 0.184i)17-s + (1.56 − 1.13i)19-s + (0.464 + 0.186i)20-s + (−0.100 + 0.699i)22-s − 0.493i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.887 + 0.460i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.887 + 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26463 - 0.308348i\)
\(L(\frac12)\) \(\approx\) \(1.26463 - 0.308348i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 \)
5 \( 1 + (-2.16 + 0.546i)T \)
11 \( 1 + (0.565 + 3.26i)T \)
good7 \( 1 + (1.82 - 2.51i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.595 - 0.193i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.33 - 0.759i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.82 + 4.96i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 2.36iT - 23T^{2} \)
29 \( 1 + (-6.27 - 4.56i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.692 + 2.13i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.60 + 6.34i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.05 + 0.769i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.80iT - 43T^{2} \)
47 \( 1 + (-3.07 - 4.22i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.77 - 0.900i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.83 + 1.33i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.46 - 13.7i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 7.21iT - 67T^{2} \)
71 \( 1 + (2.88 + 8.89i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.955 + 1.31i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.29 + 13.2i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (9.70 - 3.15i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 8.56T + 89T^{2} \)
97 \( 1 + (-5.92 - 1.92i)T + (78.4 + 57.0i)T^{2} \)
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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

−9.732048988234055589923034834108, −9.043181802619942258027933430666, −8.715716281337801452963127560258, −7.49190343985702268335177445235, −6.36240489007458988400922794806, −5.88077861340676980659600652036, −4.80360312092789101877184763482, −3.09260781460353975963264641264, −2.49088377225132479990730394629, −0.930711217242665114126705414412, 1.13216537292491067166059089035, 2.42113884325029884869687333738, 3.63916954043954081770366020940, 4.98511464283564365755522416288, 5.97484467353337336124268249109, 6.84741034758899360193031676312, 7.37263636809736290862531976657, 8.407421514422926415886539689884, 9.560985406748484919118303308449, 9.956249971144622766178682086306

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