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Properties

Label 2-990-55.4-c1-0-15
Degree $2$
Conductor $990$
Sign $0.612 + 0.790i$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−2.23 − 0.109i)5-s + (1.51 − 0.492i)7-s + (0.951 + 0.309i)8-s + (1.40 − 1.74i)10-s + (−3.31 + 0.183i)11-s + (−1.81 + 2.49i)13-s + (−0.492 + 1.51i)14-s + (−0.809 + 0.587i)16-s + (4.22 + 5.81i)17-s + (2.42 − 7.47i)19-s + (0.586 + 2.15i)20-s + (1.79 − 2.78i)22-s − 3.38i·23-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.154 − 0.475i)4-s + (−0.998 − 0.0488i)5-s + (0.573 − 0.186i)7-s + (0.336 + 0.109i)8-s + (0.443 − 0.551i)10-s + (−0.998 + 0.0554i)11-s + (−0.503 + 0.693i)13-s + (−0.131 + 0.405i)14-s + (−0.202 + 0.146i)16-s + (1.02 + 1.40i)17-s + (0.557 − 1.71i)19-s + (0.131 + 0.482i)20-s + (0.383 − 0.594i)22-s − 0.706i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.612 + 0.790i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.612 + 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663935 - 0.325694i\)
\(L(\frac12)\) \(\approx\) \(0.663935 - 0.325694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 \)
5 \( 1 + (2.23 + 0.109i)T \)
11 \( 1 + (3.31 - 0.183i)T \)
good7 \( 1 + (-1.51 + 0.492i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.81 - 2.49i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.22 - 5.81i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.42 + 7.47i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 3.38iT - 23T^{2} \)
29 \( 1 + (2.17 + 6.70i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.10 + 2.25i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.108 - 0.0352i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.35 + 10.3i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.27iT - 43T^{2} \)
47 \( 1 + (7.49 + 2.43i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.69 + 10.5i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.837 - 2.57i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.80 - 2.03i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.846iT - 67T^{2} \)
71 \( 1 + (-11.9 + 8.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-11.9 + 3.89i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.85 - 2.07i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.04 + 5.56i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + (0.957 - 1.31i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.804949541525934576390404483435, −8.817210319451733840267073862664, −8.055410825419503712761746239467, −7.51827997809292743124356635538, −6.74549277430126046452810004632, −5.47427583756933961350996549833, −4.71193332818318115891953440978, −3.73471339654240012694736066764, −2.21141183654995249351810203282, −0.44314903378999982974991487189, 1.22268092090938645615633179128, 2.85434099908813782925040047748, 3.51805054405298604860902125820, 4.89941652075530739751451299978, 5.49042174432190479015992068680, 7.19185856441069974442151584703, 7.87312402597486041260556336090, 8.158562979217137773874304860854, 9.468427647911209664937130927197, 10.08793613074984128514957155934

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