Properties

Label 2-1800-5.4-c1-0-8
Degree $2$
Conductor $1800$
Sign $0.447 - 0.894i$
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  + 5i·7-s + 6·11-s − 3i·13-s − 2i·17-s − 19-s + 2i·23-s + 6·29-s + 3·31-s + 6i·37-s − 4·41-s + 11i·43-s − 10i·47-s − 18·49-s + 8i·53-s − 6·59-s + ⋯
L(s)  = 1  + 1.88i·7-s + 1.80·11-s − 0.832i·13-s − 0.485i·17-s − 0.229·19-s + 0.417i·23-s + 1.11·29-s + 0.538·31-s + 0.986i·37-s − 0.624·41-s + 1.67i·43-s − 1.45i·47-s − 2.57·49-s + 1.09i·53-s − 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.874418057\)
\(L(\frac12)\) \(\approx\) \(1.874418057\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 5iT - 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + 3iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 11iT - 43T^{2} \)
47 \( 1 + 10iT - 47T^{2} \)
53 \( 1 - 8iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 - iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 16T + 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−9.407202015703902848825397984217, −8.595368586648690634672496436619, −8.156007475410420899399204764692, −6.79647414943752038285844377992, −6.23780407860912771200427177611, −5.43584808474449039997366784673, −4.58559624166148458569392310252, −3.34974303329519965935740777753, −2.53658007805095236408202283035, −1.30666411591999157523324950156, 0.804277555809628234372991598527, 1.80081316668870256100744924672, 3.51991588285518470925247481426, 4.09474211125503989548263842210, 4.72348164855584968301861870293, 6.31445540855114474485551391928, 6.69448113285226935091825975717, 7.40460976196614024470719546334, 8.382962759418449624942456372720, 9.169099916481333577042128737426

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