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Properties

Label 2-990-99.65-c1-0-28
Degree $2$
Conductor $990$
Sign $0.979 - 0.202i$
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 + (1.64 − 0.531i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.364 + 1.69i)6-s + (2.35 + 1.36i)7-s + 0.999·8-s + (2.43 − 1.75i)9-s + 0.999i·10-s + (−2.38 − 2.30i)11-s + (−1.28 − 1.16i)12-s + (−0.564 + 0.326i)13-s + (−2.35 + 1.36i)14-s + (1.16 − 1.28i)15-s + (−0.5 + 0.866i)16-s + 4.02·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.951 − 0.306i)3-s + (−0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.148 + 0.691i)6-s + (0.890 + 0.514i)7-s + 0.353·8-s + (0.811 − 0.583i)9-s + 0.316i·10-s + (−0.719 − 0.694i)11-s + (−0.370 − 0.335i)12-s + (−0.156 + 0.0904i)13-s + (−0.629 + 0.363i)14-s + (0.300 − 0.331i)15-s + (−0.125 + 0.216i)16-s + 0.977·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.979 - 0.202i$
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: \((2,\ 990,\ (\ :1/2),\ 0.979 - 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14238 + 0.219592i\)
\(L(\frac12)\) \(\approx\) \(2.14238 + 0.219592i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 0.531i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (2.38 + 2.30i)T \)
good7 \( 1 + (-2.35 - 1.36i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (0.564 - 0.326i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.02T + 17T^{2} \)
19 \( 1 - 4.60iT - 19T^{2} \)
23 \( 1 + (-5.00 + 2.89i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.697 - 1.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.685 - 1.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.64T + 37T^{2} \)
41 \( 1 + (2.47 + 4.28i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.92 + 4.57i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-10.2 - 5.92i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.28iT - 53T^{2} \)
59 \( 1 + (-3.30 + 1.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.88 - 3.97i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.07 - 7.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.74iT - 71T^{2} \)
73 \( 1 - 8.63iT - 73T^{2} \)
79 \( 1 + (14.1 + 8.16i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.414 + 0.717i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 15.2iT - 89T^{2} \)
97 \( 1 + (-2.76 + 4.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.909456404511962945850487225089, −8.660706135938206399240559321825, −8.574658518706731450919921482148, −7.67905665926777072257270524581, −6.85696495692400816163331750426, −5.65298189858949375610064892973, −5.05739436422764450574213914132, −3.64300752551091104244763111313, −2.40132942804608988028538144222, −1.25541687656294707340902808114, 1.41111069330288948107610768891, 2.48106477695264698033185964287, 3.38558912961803129865311685983, 4.58990879188143246814476017197, 5.23267176161332204576476287322, 7.05036299527167082883614586590, 7.60265611520481708933616585963, 8.379673930688057093497699712205, 9.274548580400442589036402888299, 9.976713471455205237223811976331

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