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Properties

Label 2-990-99.65-c1-0-15
Degree $2$
Conductor $990$
Sign $0.597 - 0.801i$
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 + (−1.72 − 0.123i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (0.971 − 1.43i)6-s + (4.54 + 2.62i)7-s + 0.999·8-s + (2.96 + 0.428i)9-s + 0.999i·10-s + (−2.48 + 2.19i)11-s + (0.756 + 1.55i)12-s + (5.39 − 3.11i)13-s + (−4.54 + 2.62i)14-s + (−1.55 + 0.756i)15-s + (−0.5 + 0.866i)16-s − 4.35·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.997 − 0.0715i)3-s + (−0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (0.396 − 0.585i)6-s + (1.71 + 0.992i)7-s + 0.353·8-s + (0.989 + 0.142i)9-s + 0.316i·10-s + (−0.749 + 0.662i)11-s + (0.218 + 0.449i)12-s + (1.49 − 0.863i)13-s + (−1.21 + 0.701i)14-s + (−0.402 + 0.195i)15-s + (−0.125 + 0.216i)16-s − 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.597 - 0.801i$
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: \((2,\ 990,\ (\ :1/2),\ 0.597 - 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12159 + 0.562766i\)
\(L(\frac12)\) \(\approx\) \(1.12159 + 0.562766i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 + 0.123i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (2.48 - 2.19i)T \)
good7 \( 1 + (-4.54 - 2.62i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (-5.39 + 3.11i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.35T + 17T^{2} \)
19 \( 1 + 0.748iT - 19T^{2} \)
23 \( 1 + (-2.87 + 1.66i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.20 + 9.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.83 - 4.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.09T + 37T^{2} \)
41 \( 1 + (-2.75 - 4.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.41 - 0.817i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.640 - 0.369i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.46iT - 53T^{2} \)
59 \( 1 + (0.877 - 0.506i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.800 - 0.462i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.821 + 1.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.56iT - 71T^{2} \)
73 \( 1 + 9.67iT - 73T^{2} \)
79 \( 1 + (-10.6 - 6.17i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.86 + 4.95i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.41iT - 89T^{2} \)
97 \( 1 + (-8.16 + 14.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.30886012203149449662541800198, −9.061357995498998135601221965900, −8.352217300256559652298866485152, −7.74342132575861361205474721544, −6.52975592407544293163140512165, −5.81712715560823687359169604307, −5.02110648416956686451260732849, −4.53157671487082653134584262909, −2.28305753450951666628854058542, −1.13316098640184454200558537112, 0.993894197963348923707342764970, 1.89145891377764071832924879605, 3.68579882040984862048114620842, 4.58130836021590344432887610887, 5.34915894361307275554729183084, 6.52784775898279779906306779458, 7.32946139167590582445318866540, 8.351064871826245774693012831451, 9.007379821132172035418934018429, 10.38671714170291062027087622725

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