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Properties

Label 2-990-165.164-c1-0-9
Degree $2$
Conductor $990$
Sign $0.135 - 0.990i$
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  + i·2-s − 4-s + (1.94 + 1.10i)5-s − 1.41·7-s − i·8-s + (−1.10 + 1.94i)10-s + (0.772 − 3.22i)11-s + 3.62·13-s − 1.41i·14-s + 16-s + 2i·17-s + 6.07i·19-s + (−1.94 − 1.10i)20-s + (3.22 + 0.772i)22-s + 7.77·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.869 + 0.493i)5-s − 0.534·7-s − 0.353i·8-s + (−0.349 + 0.614i)10-s + (0.232 − 0.972i)11-s + 1.00·13-s − 0.377i·14-s + 0.250·16-s + 0.485i·17-s + 1.39i·19-s + (−0.434 − 0.246i)20-s + (0.687 + 0.164i)22-s + 1.62·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.135 - 0.990i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.135 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31903 + 1.15090i\)
\(L(\frac12)\) \(\approx\) \(1.31903 + 1.15090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (-1.94 - 1.10i)T \)
11 \( 1 + (-0.772 + 3.22i)T \)
good7 \( 1 + 1.41T + 7T^{2} \)
13 \( 1 - 3.62T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 6.07iT - 19T^{2} \)
23 \( 1 - 7.77T + 23T^{2} \)
29 \( 1 + 5.49T + 29T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 - 1.54iT - 37T^{2} \)
41 \( 1 + 12.5T + 41T^{2} \)
43 \( 1 - 9.45T + 43T^{2} \)
47 \( 1 - 9.96T + 47T^{2} \)
53 \( 1 + 6.07T + 53T^{2} \)
59 \( 1 - 0.794iT - 59T^{2} \)
61 \( 1 - 9.96iT - 61T^{2} \)
67 \( 1 - 3.08iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 4.06T + 73T^{2} \)
79 \( 1 - 6.07iT - 79T^{2} \)
83 \( 1 + 17.3iT - 83T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−10.17261511922698605991317462867, −9.151718244549913656509494212550, −8.627836463753594033261093434496, −7.58527428901224853308439428798, −6.51718864973088966454160735368, −6.08506785781258586422649252368, −5.33408472836088781989797956903, −3.84637806582439846172504130150, −3.03722319403263689252747693418, −1.33160661465551584868133741056, 0.964644529701851235319898058659, 2.21527591351315429488550917765, 3.26143969210946318964491603142, 4.54183660195868045140526981771, 5.23748373477274100219551977376, 6.38078226566663493600883108654, 7.14749022400745842020699260506, 8.503385318929513540349736062757, 9.302407978425320225844906674955, 9.584511619502020047297077493347

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