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Properties

Label 2-990-99.65-c1-0-36
Degree $2$
Conductor $990$
Sign $-0.475 + 0.879i$
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 + (−1.21 + 1.23i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (0.462 + 1.66i)6-s + (1.10 + 0.637i)7-s − 0.999·8-s + (−0.0527 − 2.99i)9-s − 0.999i·10-s + (−2.15 − 2.51i)11-s + (1.67 + 0.433i)12-s + (−1.12 + 0.651i)13-s + (1.10 − 0.637i)14-s + (−0.433 + 1.67i)15-s + (−0.5 + 0.866i)16-s − 3.37·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.700 + 0.713i)3-s + (−0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (0.189 + 0.681i)6-s + (0.417 + 0.240i)7-s − 0.353·8-s + (−0.0175 − 0.999i)9-s − 0.316i·10-s + (−0.650 − 0.759i)11-s + (0.484 + 0.125i)12-s + (−0.313 + 0.180i)13-s + (0.295 − 0.170i)14-s + (−0.111 + 0.432i)15-s + (−0.125 + 0.216i)16-s − 0.818·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.475 + 0.879i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.475 + 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571550 - 0.958330i\)
\(L(\frac12)\) \(\approx\) \(0.571550 - 0.958330i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 - 1.23i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (2.15 + 2.51i)T \)
good7 \( 1 + (-1.10 - 0.637i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (1.12 - 0.651i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.37T + 17T^{2} \)
19 \( 1 + 5.39iT - 19T^{2} \)
23 \( 1 + (-1.82 + 1.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.37 + 4.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.73 - 3.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.60T + 37T^{2} \)
41 \( 1 + (4.86 + 8.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.29 + 4.79i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.51 + 3.76i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.11iT - 53T^{2} \)
59 \( 1 + (2.49 - 1.44i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.52 + 2.61i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.12 - 5.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + 14.1iT - 73T^{2} \)
79 \( 1 + (-5.22 - 3.01i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.809 + 1.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.53iT - 89T^{2} \)
97 \( 1 + (-5.59 + 9.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.961802455820719850195789205004, −9.029242747108172358980092588218, −8.425558655649603364528130650733, −6.87687784460424805472994104768, −6.03152080996957121828586311915, −5.03086763689068962092285108783, −4.68679832974135001868811247774, −3.36749973912944044931637051181, −2.24234469127013313475019123032, −0.49491494012457013927079045778, 1.58914061636268405186405982480, 2.84569097284587101511781130077, 4.52388038301840687129578622168, 5.08992895042462788448952954297, 6.14777575371765110329772319452, 6.71057268530132873808619767503, 7.76067278186630301914116144153, 8.058141088469211934849529059853, 9.496281581451966098900609867539, 10.32333317005138185993309921148

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