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Properties

Label 2-1800-24.11-c1-0-67
Degree $2$
Conductor $1800$
Sign $-0.816 - 0.577i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
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  − 1.41i·2-s − 2.00·4-s − 4.24i·7-s + 2.82i·8-s − 1.41i·11-s − 4.24i·13-s − 6·14-s + 4.00·16-s − 2.82i·17-s − 4·19-s − 2.00·22-s + 6·23-s − 6·26-s + 8.48i·28-s + 6·29-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s − 1.60i·7-s + 1.00i·8-s − 0.426i·11-s − 1.17i·13-s − 1.60·14-s + 1.00·16-s − 0.685i·17-s − 0.917·19-s − 0.426·22-s + 1.25·23-s − 1.17·26-s + 1.60i·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ -0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.028151449\)
\(L(\frac12)\) \(\approx\) \(1.028151449\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 4.24iT - 7T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 8.48iT - 31T^{2} \)
37 \( 1 - 4.24iT - 37T^{2} \)
41 \( 1 + 9.89iT - 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + 1.41iT - 59T^{2} \)
61 \( 1 + 8.48iT - 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 - 8.48iT - 79T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 - 7.07iT - 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.842740747636572936086464735644, −8.205361081314304704624939307082, −7.32720518326110921106398053996, −6.47535308940094793654442311750, −5.14775454467630311450647549242, −4.60229434698942712152672181359, −3.49109620217490621809330320585, −2.94560662887635769265180964005, −1.35850624110957309026676314902, −0.41031757181414092715319148318, 1.76223914003643151288611390395, 2.96411411523641432886475456508, 4.33785091652723087491702960495, 4.88939408086467286081256163084, 6.03114680424289339977062113057, 6.34248834739458434702500006255, 7.32269319567083315819448210421, 8.297745361931099178344493263400, 8.844598893443983252517936360287, 9.386751054799445554481821199265

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