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Properties

Label 2-90-5.4-c3-0-6
Degree $2$
Conductor $90$
Sign $-0.0894 + 0.995i$
Analytic cond. $5.31017$
Root an. cond. $2.30438$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 4·4-s + (−11.1 − i)5-s − 22.2i·7-s − 8i·8-s + (2 − 22.2i)10-s − 22.2·11-s − 66.8i·13-s + 44.5·14-s + 16·16-s + 62i·17-s − 84·19-s + (44.5 + 4i)20-s − 44.5i·22-s − 140i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.995 − 0.0894i)5-s − 1.20i·7-s − 0.353i·8-s + (0.0632 − 0.704i)10-s − 0.610·11-s − 1.42i·13-s + 0.850·14-s + 0.250·16-s + 0.884i·17-s − 1.01·19-s + (0.497 + 0.0447i)20-s − 0.431i·22-s − 1.26i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.0894 + 0.995i$
Analytic conductor: \(5.31017\)
Root analytic conductor: \(2.30438\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :3/2),\ -0.0894 + 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.387464 - 0.423819i\)
\(L(\frac12)\) \(\approx\) \(0.387464 - 0.423819i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 \)
5 \( 1 + (11.1 + i)T \)
good7 \( 1 + 22.2iT - 343T^{2} \)
11 \( 1 + 22.2T + 1.33e3T^{2} \)
13 \( 1 + 66.8iT - 2.19e3T^{2} \)
17 \( 1 - 62iT - 4.91e3T^{2} \)
19 \( 1 + 84T + 6.85e3T^{2} \)
23 \( 1 + 140iT - 1.21e4T^{2} \)
29 \( 1 + 200.T + 2.43e4T^{2} \)
31 \( 1 - 16T + 2.97e4T^{2} \)
37 \( 1 - 244. iT - 5.06e4T^{2} \)
41 \( 1 + 222.T + 6.89e4T^{2} \)
43 \( 1 - 356. iT - 7.95e4T^{2} \)
47 \( 1 - 100iT - 1.03e5T^{2} \)
53 \( 1 + 738iT - 1.48e5T^{2} \)
59 \( 1 - 645.T + 2.05e5T^{2} \)
61 \( 1 + 358T + 2.26e5T^{2} \)
67 \( 1 + 846. iT - 3.00e5T^{2} \)
71 \( 1 - 935.T + 3.57e5T^{2} \)
73 \( 1 + 445. iT - 3.89e5T^{2} \)
79 \( 1 - 936T + 4.93e5T^{2} \)
83 \( 1 + 1.30e3iT - 5.71e5T^{2} \)
89 \( 1 + 712.T + 7.04e5T^{2} \)
97 \( 1 - 757. iT - 9.12e5T^{2} \)
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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

−13.24398984413314448891889954343, −12.64540762533849059867298378692, −10.94948858327873715812181358188, −10.21290731422328430512076250675, −8.343039327583643301734131253441, −7.78181452922321218564237283144, −6.52556606675444621710937026135, −4.84290732846892164451490910981, −3.60719081895291658867835625412, −0.33431426395893437998791889504, 2.30800915016122662326815639252, 3.93210947567228556261447678375, 5.38256300316883885783410390837, 7.18901422582205372431469882710, 8.579563356577876791478963853548, 9.424770356612483859738832434815, 11.00675155927672649581277563692, 11.74252223632291647681187865554, 12.51208269920075805739280513516, 13.76913630671429528537914934354

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