Properties

Label 2-1800-5.4-c1-0-9
Degree $2$
Conductor $1800$
Sign $0.894 - 0.447i$
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  + 4·11-s + 6i·13-s − 6i·17-s + 4·19-s − 2·29-s − 8·31-s + 2i·37-s + 6·41-s + 12i·43-s + 8i·47-s + 7·49-s − 6i·53-s + 12·59-s + 14·61-s − 4i·67-s + ⋯
L(s)  = 1  + 1.20·11-s + 1.66i·13-s − 1.45i·17-s + 0.917·19-s − 0.371·29-s − 1.43·31-s + 0.328i·37-s + 0.937·41-s + 1.82i·43-s + 1.16i·47-s + 49-s − 0.824i·53-s + 1.56·59-s + 1.79·61-s − 0.488i·67-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
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: \((2,\ 1800,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.854215073\)
\(L(\frac12)\) \(\approx\) \(1.854215073\)
\(L(\frac{3}{2})\) 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 \( 1 - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−9.402802984915132771648375843219, −8.796132632499125503946783245918, −7.56649741167929220289857634643, −6.98845760616785178267883674835, −6.26622151735023857247601604223, −5.20999346687128674767317548152, −4.33697446811332180746825328684, −3.52669041071811231590873734600, −2.28470657279009699079811902363, −1.12188486750357221554100410567, 0.858435137519986371005479529244, 2.12072137395380159076555044239, 3.50104030966001246223195561515, 3.96417219941928330908910833748, 5.44699920865496374295878712759, 5.77910362268210600002936997576, 6.93918352503521950128550326838, 7.59962791594882709441493335944, 8.528587581033797806845045095530, 9.094743344919859877060610482073

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