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Properties

Label 2-990-9.4-c1-0-9
Degree $2$
Conductor $990$
Sign $-0.701 - 0.713i$
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  + (−0.5 − 0.866i)2-s + (0.513 + 1.65i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (1.17 − 1.27i)6-s + (2.32 + 4.02i)7-s + 0.999·8-s + (−2.47 + 1.69i)9-s + 0.999·10-s + (−0.5 − 0.866i)11-s + (−1.68 − 0.382i)12-s + (−2.26 + 3.92i)13-s + (2.32 − 4.02i)14-s + (−1.68 − 0.382i)15-s + (−0.5 − 0.866i)16-s − 6.45·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.296 + 0.954i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.479 − 0.519i)6-s + (0.877 + 1.52i)7-s + 0.353·8-s + (−0.824 + 0.566i)9-s + 0.316·10-s + (−0.150 − 0.261i)11-s + (−0.487 − 0.110i)12-s + (−0.628 + 1.08i)13-s + (0.620 − 1.07i)14-s + (−0.436 − 0.0986i)15-s + (−0.125 − 0.216i)16-s − 1.56·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.701 - 0.713i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (661, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.701 - 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.423717 + 1.01071i\)
\(L(\frac12)\) \(\approx\) \(0.423717 + 1.01071i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.513 - 1.65i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-2.32 - 4.02i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (2.26 - 3.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.45T + 17T^{2} \)
19 \( 1 - 7.32T + 19T^{2} \)
23 \( 1 + (-0.277 + 0.481i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.658 + 1.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.76 + 6.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.35T + 37T^{2} \)
41 \( 1 + (-0.989 + 1.71i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.18 + 5.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.735 - 1.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.77T + 53T^{2} \)
59 \( 1 + (4.37 - 7.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.743 - 1.28i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.40 - 12.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.84T + 71T^{2} \)
73 \( 1 - 1.09T + 73T^{2} \)
79 \( 1 + (-3.92 - 6.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.75 - 3.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (-3.97 - 6.87i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.23649146910086303534271191334, −9.354045736568847158211198363234, −8.886518970561629343554828125681, −8.187655413183310234266854441504, −7.13848526865838574481253924624, −5.73133938588463587295681677611, −4.89392487322527783426977797867, −4.04197132896045967734353032214, −2.73404531648822662631369235729, −2.13966512278636021599988641096, 0.54803227193429155928139697957, 1.61882724245877564374595691118, 3.24126706693116434301414363568, 4.61573597382185290302585782547, 5.30596551331293619153818082986, 6.69294355230847829866634668015, 7.30317596439601475899909941369, 7.85638720092165343132592934820, 8.502813536477692046585791650278, 9.517254647294296446641981055423

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