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Properties

Label 2-990-55.32-c1-0-23
Degree $2$
Conductor $990$
Sign $-0.838 + 0.545i$
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 + (0.707 − 2.12i)5-s + (−2.12 + 2.12i)7-s + (0.707 + 0.707i)8-s + (0.999 + 2i)10-s + (1.41 − 3i)11-s + (−3 − 3i)13-s − 3i·14-s − 1.00·16-s + (−5.12 + 5.12i)17-s + 3·19-s + (−2.12 − 0.707i)20-s + (1.12 + 3.12i)22-s + (0.171 − 0.171i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.316 − 0.948i)5-s + (−0.801 + 0.801i)7-s + (0.250 + 0.250i)8-s + (0.316 + 0.632i)10-s + (0.426 − 0.904i)11-s + (−0.832 − 0.832i)13-s − 0.801i·14-s − 0.250·16-s + (−1.24 + 1.24i)17-s + 0.688·19-s + (−0.474 − 0.158i)20-s + (0.239 + 0.665i)22-s + (0.0357 − 0.0357i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.838 + 0.545i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.838 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0724969 - 0.244407i\)
\(L(\frac12)\) \(\approx\) \(0.0724969 - 0.244407i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 2.12i)T \)
11 \( 1 + (-1.41 + 3i)T \)
good7 \( 1 + (2.12 - 2.12i)T - 7iT^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (5.12 - 5.12i)T - 17iT^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + (-0.171 + 0.171i)T - 23iT^{2} \)
29 \( 1 + 1.24T + 29T^{2} \)
31 \( 1 + 7.24T + 31T^{2} \)
37 \( 1 + (-0.121 - 0.121i)T + 37iT^{2} \)
41 \( 1 + 1.75iT - 41T^{2} \)
43 \( 1 + (1.24 + 1.24i)T + 43iT^{2} \)
47 \( 1 + (4.41 + 4.41i)T + 47iT^{2} \)
53 \( 1 + (9.53 - 9.53i)T - 53iT^{2} \)
59 \( 1 + 1.41iT - 59T^{2} \)
61 \( 1 - 7.24iT - 61T^{2} \)
67 \( 1 + (4 + 4i)T + 67iT^{2} \)
71 \( 1 - 1.24T + 71T^{2} \)
73 \( 1 + (6 + 6i)T + 73iT^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + (7.24 + 7.24i)T + 83iT^{2} \)
89 \( 1 + 5.48iT - 89T^{2} \)
97 \( 1 + (2.24 + 2.24i)T + 97iT^{2} \)
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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

−9.368361762726151832019126848624, −8.876410798855578327196864980152, −8.216094830579317962400973741648, −7.14511086555267631416372645790, −5.97551903719616258380223104972, −5.70859126105264996977634910836, −4.52911004918519246591801350121, −3.16055429791041915589656037655, −1.77275148522692929384719701729, −0.12965642729069762410885000922, 1.86367778932178693395655009509, 2.87277926157484832720126491544, 3.89574342940647732866082279220, 4.91621710190037144710798618331, 6.51816296474292181509996319272, 7.03875008288551431915457889067, 7.53432999414304979066887316585, 9.110825616721331033141590773952, 9.658413398323979120935557675098, 10.06814874089692175827143940935

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