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Properties

Label 2-990-55.43-c1-0-23
Degree $2$
Conductor $990$
Sign $0.975 + 0.221i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (1.22 − 1.87i)5-s + (−0.645 − 0.645i)7-s + (−0.707 + 0.707i)8-s + (2.18 − 0.456i)10-s + (−3.29 + 0.414i)11-s + (4.97 − 4.97i)13-s − 0.912i·14-s − 1.00·16-s + (2.29 + 2.29i)17-s + 4.82·19-s + (1.87 + 1.22i)20-s + (−2.61 − 2.03i)22-s + (−3.14 − 3.14i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.547 − 0.836i)5-s + (−0.243 − 0.243i)7-s + (−0.250 + 0.250i)8-s + (0.692 − 0.144i)10-s + (−0.992 + 0.124i)11-s + (1.37 − 1.37i)13-s − 0.243i·14-s − 0.250·16-s + (0.555 + 0.555i)17-s + 1.10·19-s + (0.418 + 0.273i)20-s + (−0.558 − 0.433i)22-s + (−0.656 − 0.656i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.975 + 0.221i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.975 + 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21142 - 0.247424i\)
\(L(\frac12)\) \(\approx\) \(2.21142 - 0.247424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-1.22 + 1.87i)T \)
11 \( 1 + (3.29 - 0.414i)T \)
good7 \( 1 + (0.645 + 0.645i)T + 7iT^{2} \)
13 \( 1 + (-4.97 + 4.97i)T - 13iT^{2} \)
17 \( 1 + (-2.29 - 2.29i)T + 17iT^{2} \)
19 \( 1 - 4.82T + 19T^{2} \)
23 \( 1 + (3.14 + 3.14i)T + 23iT^{2} \)
29 \( 1 - 3.92T + 29T^{2} \)
31 \( 1 - 4.65T + 31T^{2} \)
37 \( 1 + (-3.41 + 3.41i)T - 37iT^{2} \)
41 \( 1 + 3.34iT - 41T^{2} \)
43 \( 1 + (-1.04 + 1.04i)T - 43iT^{2} \)
47 \( 1 + (3.88 - 3.88i)T - 47iT^{2} \)
53 \( 1 + (-1.81 - 1.81i)T + 53iT^{2} \)
59 \( 1 + 2.99iT - 59T^{2} \)
61 \( 1 - 14.7iT - 61T^{2} \)
67 \( 1 + (7.70 - 7.70i)T - 67iT^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + (-7.29 + 7.29i)T - 73iT^{2} \)
79 \( 1 + 5.91T + 79T^{2} \)
83 \( 1 + (-6.14 + 6.14i)T - 83iT^{2} \)
89 \( 1 - 0.726iT - 89T^{2} \)
97 \( 1 + (3.75 - 3.75i)T - 97iT^{2} \)
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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

−10.16347816145409969030373037557, −8.905208275970905054526383451951, −8.174508575060453581095321632186, −7.59719943621718180658578555679, −6.19910220009732132008903498816, −5.71043131126748742684547197603, −4.89524982815279797131979721859, −3.77101221269540663301252597909, −2.70821177870632969896556168755, −0.975042705947011458359065949049, 1.51052737622203350367999591850, 2.75796641704463493094393953418, 3.46829591661015962982138562343, 4.73049359729419171041634874159, 5.79456519800338599466082248531, 6.36150511799207653077621203400, 7.35550104394947288272196752762, 8.423737810633580493048020382839, 9.605258809898480849044804895056, 9.927685546409783636914250769679

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