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Properties

Label 2-990-55.32-c1-0-21
Degree $2$
Conductor $990$
Sign $-0.252 + 0.967i$
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 + (−2 − i)5-s + (2.82 − 2.82i)7-s + (0.707 + 0.707i)8-s + (2.12 − 0.707i)10-s + (3 − 1.41i)11-s + (−1.41 − 1.41i)13-s + 4.00i·14-s − 1.00·16-s + (1.41 − 1.41i)17-s − 8.48·19-s + (−1.00 + 2.00i)20-s + (−1.12 + 3.12i)22-s + (−3 + 3i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.894 − 0.447i)5-s + (1.06 − 1.06i)7-s + (0.250 + 0.250i)8-s + (0.670 − 0.223i)10-s + (0.904 − 0.426i)11-s + (−0.392 − 0.392i)13-s + 1.06i·14-s − 0.250·16-s + (0.342 − 0.342i)17-s − 1.94·19-s + (−0.223 + 0.447i)20-s + (−0.239 + 0.665i)22-s + (−0.625 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.252 + 0.967i$
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: \((2,\ 990,\ (\ :1/2),\ -0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.483431 - 0.625579i\)
\(L(\frac12)\) \(\approx\) \(0.483431 - 0.625579i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
11 \( 1 + (-3 + 1.41i)T \)
good7 \( 1 + (-2.82 + 2.82i)T - 7iT^{2} \)
13 \( 1 + (1.41 + 1.41i)T + 13iT^{2} \)
17 \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \)
19 \( 1 + 8.48T + 19T^{2} \)
23 \( 1 + (3 - 3i)T - 23iT^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 + (2.82 + 2.82i)T + 43iT^{2} \)
47 \( 1 + (-1 - i)T + 47iT^{2} \)
53 \( 1 + (3 - 3i)T - 53iT^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + 14.1iT - 61T^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-7.07 - 7.07i)T + 73iT^{2} \)
79 \( 1 - 2.82T + 79T^{2} \)
83 \( 1 + (5.65 + 5.65i)T + 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (7 + 7i)T + 97iT^{2} \)
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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

−9.632187444787522422329821203491, −8.633806143381856581767802941450, −8.093184278999007215459047017437, −7.40749097854714316018817545637, −6.59744012675904812153380726912, −5.36502973345812545763763445032, −4.41312005889468698818284349999, −3.73405418666427563633014942726, −1.72553210939248857473373519835, −0.43844413472104638530493798216, 1.70370220137157879971421784018, 2.63491077602132279163787352291, 4.05294688970705792632627900964, 4.63894298676932104818617427139, 6.12266888172147421944000213131, 6.97774526383859681997628035302, 8.090319190563441111839385711761, 8.453261600830740120214614106838, 9.314268378269628627272409130033, 10.36588495649345980535170210679

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