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Properties

Label 2-990-33.2-c1-0-7
Degree $2$
Conductor $990$
Sign $0.943 + 0.330i$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (−1.76 + 2.42i)7-s + (−0.809 + 0.587i)8-s − 0.999i·10-s + (3.31 − 0.0399i)11-s + (2.08 + 0.677i)13-s + (1.76 + 2.42i)14-s + (0.309 + 0.951i)16-s + (0.0911 + 0.280i)17-s + (0.0850 + 0.117i)19-s + (−0.951 − 0.309i)20-s + (0.986 − 3.16i)22-s + 5.49i·23-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (0.425 − 0.138i)5-s + (−0.666 + 0.917i)7-s + (−0.286 + 0.207i)8-s − 0.316i·10-s + (0.999 − 0.0120i)11-s + (0.578 + 0.187i)13-s + (0.471 + 0.649i)14-s + (0.0772 + 0.237i)16-s + (0.0220 + 0.0680i)17-s + (0.0195 + 0.0268i)19-s + (−0.212 − 0.0690i)20-s + (0.210 − 0.675i)22-s + 1.14i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.943 + 0.330i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80054 - 0.306235i\)
\(L(\frac12)\) \(\approx\) \(1.80054 - 0.306235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 \)
5 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (-3.31 + 0.0399i)T \)
good7 \( 1 + (1.76 - 2.42i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-2.08 - 0.677i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.0911 - 0.280i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.0850 - 0.117i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 5.49iT - 23T^{2} \)
29 \( 1 + (-6.19 - 4.50i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.47 + 4.54i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.38 - 2.45i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-8.45 + 6.14i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + (-6.59 - 9.07i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.69 + 0.551i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.12 - 5.68i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.02 - 0.656i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 + (-2.61 + 0.850i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.581 + 0.800i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (9.46 + 3.07i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.272 + 0.838i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 3.26iT - 89T^{2} \)
97 \( 1 + (-2.85 + 8.78i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.891641144134105661773756400963, −9.087648307370202112605781646568, −8.806196565588726345467038102307, −7.37948393906366349755767421141, −6.15954285376968257949689779763, −5.79337747428982469082651149884, −4.51655835829586787789942673329, −3.51355824276015848112283556717, −2.50325443153944524251899064072, −1.26811465229902572556741658193, 0.964408228622138612036169496733, 2.84777973481974953815066171450, 3.94178282710056424732229210792, 4.69728153374339915243371072272, 6.16635451279717509747489604136, 6.41518453768967035280919870175, 7.34413767397684139767429428689, 8.312824116482126542709533512168, 9.172561727499824996094947456426, 9.956078104825042181702621677199

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