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Properties

Label 2-990-99.65-c1-0-7
Degree $2$
Conductor $990$
Sign $0.429 - 0.902i$
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 + (−1.19 − 1.25i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (1.68 − 0.407i)6-s + (−1.93 − 1.11i)7-s + 0.999·8-s + (−0.147 + 2.99i)9-s + 0.999i·10-s + (2.43 + 2.24i)11-s + (−0.489 + 1.66i)12-s + (−2.70 + 1.55i)13-s + (1.93 − 1.11i)14-s + (−1.66 − 0.489i)15-s + (−0.5 + 0.866i)16-s − 3.53·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.689 − 0.724i)3-s + (−0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (0.687 − 0.166i)6-s + (−0.732 − 0.422i)7-s + 0.353·8-s + (−0.0490 + 0.998i)9-s + 0.316i·10-s + (0.735 + 0.677i)11-s + (−0.141 + 0.479i)12-s + (−0.749 + 0.432i)13-s + (0.517 − 0.299i)14-s + (−0.429 − 0.126i)15-s + (−0.125 + 0.216i)16-s − 0.856·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.429 - 0.902i$
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: \((2,\ 990,\ (\ :1/2),\ 0.429 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.655911 + 0.414233i\)
\(L(\frac12)\) \(\approx\) \(0.655911 + 0.414233i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 1.25i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-2.43 - 2.24i)T \)
good7 \( 1 + (1.93 + 1.11i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (2.70 - 1.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.53T + 17T^{2} \)
19 \( 1 - 0.649iT - 19T^{2} \)
23 \( 1 + (-1.72 + 0.997i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0531 + 0.0920i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.17 - 7.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.00T + 37T^{2} \)
41 \( 1 + (-1.38 - 2.39i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-10.0 - 5.78i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.60 - 3.23i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.95iT - 53T^{2} \)
59 \( 1 + (2.80 - 1.61i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.4 - 6.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.91 - 8.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.16iT - 71T^{2} \)
73 \( 1 + 1.40iT - 73T^{2} \)
79 \( 1 + (-2.59 - 1.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.91 - 8.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.269iT - 89T^{2} \)
97 \( 1 + (2.53 - 4.38i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.01648244430734807750544433409, −9.317378780624960927516561939440, −8.427141230011525525995062849444, −7.25284665849183288724455844433, −6.82606467314773454405994191948, −6.17105894147275231499882575859, −5.08055176024090880420917463266, −4.25425676278669303518225162938, −2.39987153946358631411170631773, −1.09385554121221001997793612461, 0.52367366845935579879480022432, 2.41890171346770918082660770119, 3.43479129743371790543194288403, 4.40681678243454476094410417132, 5.55449604871629840114005369698, 6.28483149113532513531790664856, 7.19822351260553360614920174724, 8.601415320216219853920245664138, 9.324939485621334545640720934789, 9.782582690565645464199460305837

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