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Properties

Label 2-990-99.32-c1-0-0
Degree $2$
Conductor $990$
Sign $-0.999 - 0.0397i$
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 + (−0.152 − 1.72i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (1.41 − 0.994i)6-s + (−0.0947 + 0.0546i)7-s − 0.999·8-s + (−2.95 + 0.525i)9-s − 0.999i·10-s + (−3.28 + 0.440i)11-s + (1.57 + 0.730i)12-s + (4.34 + 2.50i)13-s + (−0.0947 − 0.0546i)14-s + (−0.730 + 1.57i)15-s + (−0.5 − 0.866i)16-s − 5.43·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.0879 − 0.996i)3-s + (−0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.578 − 0.406i)6-s + (−0.0358 + 0.0206i)7-s − 0.353·8-s + (−0.984 + 0.175i)9-s − 0.316i·10-s + (−0.991 + 0.132i)11-s + (0.453 + 0.210i)12-s + (1.20 + 0.695i)13-s + (−0.0253 − 0.0146i)14-s + (−0.188 + 0.405i)15-s + (−0.125 − 0.216i)16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.999 - 0.0397i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.999 - 0.0397i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000729249 + 0.0366570i\)
\(L(\frac12)\) \(\approx\) \(0.000729249 + 0.0366570i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.152 + 1.72i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (3.28 - 0.440i)T \)
good7 \( 1 + (0.0947 - 0.0546i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (-4.34 - 2.50i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.43T + 17T^{2} \)
19 \( 1 + 3.92iT - 19T^{2} \)
23 \( 1 + (2.33 + 1.34i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.51 + 6.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.17 - 7.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + (4.65 - 8.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.78 - 2.76i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.66 + 2.11i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.2iT - 53T^{2} \)
59 \( 1 + (-3.93 - 2.27i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.64 + 4.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.459 + 0.795i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + 5.22iT - 73T^{2} \)
79 \( 1 + (-0.0924 + 0.0533i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.41 + 12.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.300iT - 89T^{2} \)
97 \( 1 + (-1.17 - 2.02i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.64773713464003474940555557244, −9.108489308442677420220197936597, −8.555070237610161216584657273703, −7.80767451907839920233705634894, −6.90412891120790753774179928360, −6.35556654667743481694426138628, −5.32063083629409109776382805443, −4.40538467502273275978091155981, −3.15121164857831731867224160852, −1.86513478493570066886604522797, 0.01411819964988281404340044289, 2.17319114830496978768591886918, 3.53548155790002198190747658249, 3.84557461907512763564495888364, 5.22020467983864716900355616402, 5.69010353791404369927927133585, 6.93421524383691369551167502036, 8.305986939579667242891225318147, 8.722558847718891235636114083000, 9.922080256978519520417630213844

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