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Properties

Label 2-990-495.164-c1-0-67
Degree $2$
Conductor $990$
Sign $-0.736 - 0.675i$
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  + (−0.866 − 0.5i)2-s + (−0.690 − 1.58i)3-s + (0.499 + 0.866i)4-s + (−1.04 − 1.97i)5-s + (−0.196 + 1.72i)6-s + (0.998 − 1.72i)7-s − 0.999i·8-s + (−2.04 + 2.19i)9-s + (−0.0827 + 2.23i)10-s + (0.989 − 3.16i)11-s + (1.03 − 1.39i)12-s + (−0.497 − 0.862i)13-s + (−1.72 + 0.998i)14-s + (−2.41 + 3.02i)15-s + (−0.5 + 0.866i)16-s − 0.515i·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.398 − 0.917i)3-s + (0.249 + 0.433i)4-s + (−0.467 − 0.883i)5-s + (−0.0802 + 0.702i)6-s + (0.377 − 0.653i)7-s − 0.353i·8-s + (−0.682 + 0.730i)9-s + (−0.0261 + 0.706i)10-s + (0.298 − 0.954i)11-s + (0.297 − 0.401i)12-s + (−0.138 − 0.239i)13-s + (−0.462 + 0.266i)14-s + (−0.624 + 0.781i)15-s + (−0.125 + 0.216i)16-s − 0.125i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.736 - 0.675i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.736 - 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.224509 + 0.576911i\)
\(L(\frac12)\) \(\approx\) \(0.224509 + 0.576911i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.690 + 1.58i)T \)
5 \( 1 + (1.04 + 1.97i)T \)
11 \( 1 + (-0.989 + 3.16i)T \)
good7 \( 1 + (-0.998 + 1.72i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (0.497 + 0.862i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.515iT - 17T^{2} \)
19 \( 1 + 3.42iT - 19T^{2} \)
23 \( 1 + (2.97 + 5.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.19 + 3.80i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.0604 - 0.104i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.46iT - 37T^{2} \)
41 \( 1 + (2.08 + 3.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0818 + 0.141i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.478 - 0.828i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.55T + 53T^{2} \)
59 \( 1 + (3.95 - 2.28i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.10 - 2.94i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.990 - 0.571i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.982iT - 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + (-6.91 - 3.99i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.9 - 6.34i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.77iT - 89T^{2} \)
97 \( 1 + (0.229 + 0.132i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.338332323507096063426116846869, −8.363899148650856985005943703670, −8.075046005198382465856260146598, −7.10157545046676521595709087690, −6.23171783948023998914673388680, −5.09063761430466479145723180354, −4.09397698065821087121385295368, −2.70218336975114648946194242942, −1.27124878991704046166110121011, −0.40123690003804736415630150406, 1.97677566357054672172462399580, 3.37404214467189666482671149134, 4.38982253216873926139351414358, 5.44140325466880023930767836466, 6.26912763467631307274667186156, 7.17888278781146322898049180123, 8.027456465852615740556100148189, 8.961445712462503497671343798623, 9.756526543519564248412738316016, 10.32942256859387064386250407205

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