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Properties

Label 2-990-15.2-c1-0-8
Degree $2$
Conductor $990$
Sign $0.859 + 0.511i$
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 + (−1.77 + 1.35i)5-s + (−0.425 − 0.425i)7-s + (0.707 + 0.707i)8-s + (0.301 − 2.21i)10-s − i·11-s + (0.719 − 0.719i)13-s + 0.602·14-s − 1.00·16-s + (−2.48 + 2.48i)17-s − 5.57i·19-s + (1.35 + 1.77i)20-s + (0.707 + 0.707i)22-s + (−0.605 − 0.605i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.795 + 0.605i)5-s + (−0.160 − 0.160i)7-s + (0.250 + 0.250i)8-s + (0.0952 − 0.700i)10-s − 0.301i·11-s + (0.199 − 0.199i)13-s + 0.160·14-s − 0.250·16-s + (−0.603 + 0.603i)17-s − 1.27i·19-s + (0.302 + 0.397i)20-s + (0.150 + 0.150i)22-s + (−0.126 − 0.126i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.760551 - 0.209175i\)
\(L(\frac12)\) \(\approx\) \(0.760551 - 0.209175i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.77 - 1.35i)T \)
11 \( 1 + iT \)
good7 \( 1 + (0.425 + 0.425i)T + 7iT^{2} \)
13 \( 1 + (-0.719 + 0.719i)T - 13iT^{2} \)
17 \( 1 + (2.48 - 2.48i)T - 17iT^{2} \)
19 \( 1 + 5.57iT - 19T^{2} \)
23 \( 1 + (0.605 + 0.605i)T + 23iT^{2} \)
29 \( 1 - 2.10T + 29T^{2} \)
31 \( 1 - 1.21T + 31T^{2} \)
37 \( 1 + (-1.39 - 1.39i)T + 37iT^{2} \)
41 \( 1 + 1.73iT - 41T^{2} \)
43 \( 1 + (-1.80 + 1.80i)T - 43iT^{2} \)
47 \( 1 + (-9.34 + 9.34i)T - 47iT^{2} \)
53 \( 1 + (-1.63 - 1.63i)T + 53iT^{2} \)
59 \( 1 - 3.88T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + (-0.179 - 0.179i)T + 67iT^{2} \)
71 \( 1 + 6.06iT - 71T^{2} \)
73 \( 1 + (-0.0583 + 0.0583i)T - 73iT^{2} \)
79 \( 1 + 10.8iT - 79T^{2} \)
83 \( 1 + (4.03 + 4.03i)T + 83iT^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 + (-5.25 - 5.25i)T + 97iT^{2} \)
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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

−9.975727580014486167661462098513, −8.793479220328677906354349006357, −8.360744554109229731407998467528, −7.27336083886952573991977398151, −6.78522422684669604575383535590, −5.83995865907526086022517281512, −4.63381019803614629096665512269, −3.64416973128631983225432200524, −2.42200906469185177122596853270, −0.50981988281906613942955267169, 1.12657049896518670146797265346, 2.54926616173796264244701791009, 3.79624492721406913005634670894, 4.52576789966430772738378053206, 5.71808610587504770396438667133, 6.91168682752253986552218025044, 7.77377268660530284703983311648, 8.459123287233877126478172368345, 9.250061754911363176520032251890, 9.948817060483863991302916536684

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