Properties

Label 2-90-1.1-c1-0-2
Degree $2$
Conductor $90$
Sign $1$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s + 2·13-s − 4·14-s + 16-s − 6·17-s − 4·19-s + 20-s + 25-s + 2·26-s − 4·28-s + 6·29-s + 8·31-s + 32-s − 6·34-s − 4·35-s + 2·37-s − 4·38-s + 40-s + 6·41-s − 4·43-s + 9·49-s + 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.676·35-s + 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + 9/7·49-s + 0.141·50-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.337595994\)
\(L(\frac12)\) \(\approx\) \(1.337595994\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.78806788970128105043488414911, −13.19720570989419308005927819605, −12.30986793353697911256504493943, −10.91913531404329383721387224500, −9.880541566397139490214892173475, −8.644229019078172602684577487278, −6.73106470746564203527567325446, −6.11397905422548756742635703073, −4.32687879913691587957737883725, −2.74628747875934588873509462774, 2.74628747875934588873509462774, 4.32687879913691587957737883725, 6.11397905422548756742635703073, 6.73106470746564203527567325446, 8.644229019078172602684577487278, 9.880541566397139490214892173475, 10.91913531404329383721387224500, 12.30986793353697911256504493943, 13.19720570989419308005927819605, 13.78806788970128105043488414911

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