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Properties

Label 2-990-99.32-c1-0-39
Degree $2$
Conductor $990$
Sign $0.923 + 0.382i$
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 + (−0.805 + 1.53i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (−1.73 + 0.0689i)6-s + (1.00 − 0.582i)7-s − 0.999·8-s + (−1.70 − 2.47i)9-s − 0.999i·10-s + (1.99 − 2.64i)11-s + (−0.925 − 1.46i)12-s + (−4.84 − 2.79i)13-s + (1.00 + 0.582i)14-s + (1.46 − 0.925i)15-s + (−0.5 − 0.866i)16-s + 4.27·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.465 + 0.885i)3-s + (−0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (−0.706 + 0.0281i)6-s + (0.381 − 0.220i)7-s − 0.353·8-s + (−0.567 − 0.823i)9-s − 0.316i·10-s + (0.602 − 0.798i)11-s + (−0.267 − 0.422i)12-s + (−1.34 − 0.775i)13-s + (0.269 + 0.155i)14-s + (0.378 − 0.238i)15-s + (−0.125 − 0.216i)16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.923 + 0.382i$
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: \((2,\ 990,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.999060 - 0.198590i\)
\(L(\frac12)\) \(\approx\) \(0.999060 - 0.198590i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.805 - 1.53i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-1.99 + 2.64i)T \)
good7 \( 1 + (-1.00 + 0.582i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (4.84 + 2.79i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.27T + 17T^{2} \)
19 \( 1 + 2.48iT - 19T^{2} \)
23 \( 1 + (5.58 + 3.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.38 + 5.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.622 + 1.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.55T + 37T^{2} \)
41 \( 1 + (0.777 - 1.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.2 + 5.92i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.27 - 4.77i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.97iT - 53T^{2} \)
59 \( 1 + (0.198 + 0.114i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.03 + 4.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.13 - 12.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.4iT - 71T^{2} \)
73 \( 1 - 3.94iT - 73T^{2} \)
79 \( 1 + (-9.95 + 5.74i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.15 + 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.0iT - 89T^{2} \)
97 \( 1 + (-5.52 - 9.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.872334840740044956299791713227, −9.153551986948863990127237533934, −8.100690513616502581817890048384, −7.53975424946256533243505462037, −6.24980572778662242581871638468, −5.58474731881582433401035445014, −4.67020973976599379132397692945, −3.97908610366151248936217799938, −2.89218234488852710665247545990, −0.45486647990857112568884577818, 1.48523700825992616556015020643, 2.35008774610588048458946186282, 3.74022940599628272597551833921, 4.82315320554837284924789882311, 5.61418137099249349669528332742, 6.70536722842488301706662651899, 7.44133401576661418000158513398, 8.186170777941804588324650155883, 9.472596408839289896614134153359, 10.07313295864533499763510337461

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