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Properties

Label 2-990-99.32-c1-0-36
Degree $2$
Conductor $990$
Sign $0.999 - 0.00434i$
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 + (0.216 + 1.71i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.5i)5-s + (1.38 − 1.04i)6-s + (3.44 − 1.98i)7-s + 0.999·8-s + (−2.90 + 0.743i)9-s − 0.999i·10-s + (3.29 − 0.340i)11-s + (−1.59 − 0.671i)12-s + (1.91 + 1.10i)13-s + (−3.44 − 1.98i)14-s + (−0.671 + 1.59i)15-s + (−0.5 − 0.866i)16-s + 2.92·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.124 + 0.992i)3-s + (−0.249 + 0.433i)4-s + (0.387 + 0.223i)5-s + (0.563 − 0.427i)6-s + (1.30 − 0.751i)7-s + 0.353·8-s + (−0.968 + 0.247i)9-s − 0.316i·10-s + (0.994 − 0.102i)11-s + (−0.460 − 0.193i)12-s + (0.531 + 0.306i)13-s + (−0.920 − 0.531i)14-s + (−0.173 + 0.412i)15-s + (−0.125 − 0.216i)16-s + 0.709·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.999 - 0.00434i$
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: \((2,\ 990,\ (\ :1/2),\ 0.999 - 0.00434i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76705 + 0.00384306i\)
\(L(\frac12)\) \(\approx\) \(1.76705 + 0.00384306i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.216 - 1.71i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-3.29 + 0.340i)T \)
good7 \( 1 + (-3.44 + 1.98i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (-1.91 - 1.10i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.92T + 17T^{2} \)
19 \( 1 + 5.09iT - 19T^{2} \)
23 \( 1 + (3.45 + 1.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.94 + 3.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.189 - 0.328i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.03T + 37T^{2} \)
41 \( 1 + (2.82 - 4.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.34 + 3.08i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.07 - 4.08i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.04iT - 53T^{2} \)
59 \( 1 + (-11.5 - 6.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.64 + 1.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.41 - 4.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 - 1.49iT - 73T^{2} \)
79 \( 1 + (-9.26 + 5.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.29 - 5.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.0iT - 89T^{2} \)
97 \( 1 + (6.50 + 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.00450317756276712525857325925, −9.334042356571365891814533042076, −8.518601431691238281238399703893, −7.80152750265560896896471504436, −6.61884680397968475679278851818, −5.42373450725856472987681775096, −4.38701858540630478265960417580, −3.84655834592243049715102456656, −2.52509442223493851594722979661, −1.19544036590327872448154404490, 1.28657726555047212665321236276, 1.98515742986192374195187451596, 3.69832222094325342890816629219, 5.19390441490874702552509085024, 5.80085121932824795372604015537, 6.56750969622896969609688273251, 7.72048778901206280631781827871, 8.172868768978237668573039037625, 8.883527626721641247654592819805, 9.690519114184948075272589133448

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