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Properties

Label 2-990-99.65-c1-0-32
Degree $2$
Conductor $990$
Sign $-0.394 + 0.918i$
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 + (−1.28 − 1.16i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (1.65 − 0.526i)6-s + (−0.700 − 0.404i)7-s + 0.999·8-s + (0.281 + 2.98i)9-s + 0.999i·10-s + (−2.19 − 2.48i)11-s + (−0.369 + 1.69i)12-s + (1.59 − 0.918i)13-s + (0.700 − 0.404i)14-s + (−1.69 − 0.369i)15-s + (−0.5 + 0.866i)16-s + 6.50·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.739 − 0.673i)3-s + (−0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (0.673 − 0.214i)6-s + (−0.264 − 0.152i)7-s + 0.353·8-s + (0.0937 + 0.995i)9-s + 0.316i·10-s + (−0.660 − 0.750i)11-s + (−0.106 + 0.488i)12-s + (0.441 − 0.254i)13-s + (0.187 − 0.108i)14-s + (−0.436 − 0.0953i)15-s + (−0.125 + 0.216i)16-s + 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.394 + 0.918i$
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: \((2,\ 990,\ (\ :1/2),\ -0.394 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350769 - 0.532616i\)
\(L(\frac12)\) \(\approx\) \(0.350769 - 0.532616i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 + 1.16i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (2.19 + 2.48i)T \)
good7 \( 1 + (0.700 + 0.404i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (-1.59 + 0.918i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.50T + 17T^{2} \)
19 \( 1 + 1.87iT - 19T^{2} \)
23 \( 1 + (4.50 - 2.59i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0530 - 0.0918i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.25 + 5.64i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.15T + 37T^{2} \)
41 \( 1 + (1.74 + 3.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.09 - 0.633i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.05 + 2.34i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 + (1.38 - 0.799i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.7 + 6.21i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.221 + 0.383i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.87iT - 71T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 + (10.8 + 6.23i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.77 - 9.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.20iT - 89T^{2} \)
97 \( 1 + (-5.63 + 9.75i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.813495139311793846615050002815, −8.712877841692825878454083982406, −7.84270139353973036375193847442, −7.33089889867602346855338958694, −6.02998596608808254645337127831, −5.82775170608921092089002775499, −4.85888941712901363767243764730, −3.31240449292892257166205630721, −1.69625854095666670242843448009, −0.37815139314152038717322318032, 1.50321734381461397016867619519, 2.97910280254873589219961605950, 3.91818309332950590783402093844, 5.03272179532089697762479622292, 5.81948340373603516094153361206, 6.79969668507730786059845583344, 7.85998211379855015465281359817, 8.855599500866077413458537400974, 9.909410182644028306899304424813, 10.06073521830756853627545899270

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