Properties

Label 2-1800-5.4-c3-0-66
Degree $2$
Conductor $1800$
Sign $-0.894 - 0.447i$
Analytic cond. $106.203$
Root an. cond. $10.3055$
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  − 34i·7-s − 16·11-s − 58i·13-s + 70i·17-s − 4·19-s − 134i·23-s − 242·29-s + 100·31-s − 438i·37-s + 138·41-s − 178i·43-s − 22i·47-s − 813·49-s + 162i·53-s − 268·59-s + ⋯
L(s)  = 1  − 1.83i·7-s − 0.438·11-s − 1.23i·13-s + 0.998i·17-s − 0.0482·19-s − 1.21i·23-s − 1.54·29-s + 0.579·31-s − 1.94i·37-s + 0.525·41-s − 0.631i·43-s − 0.0682i·47-s − 2.37·49-s + 0.419i·53-s − 0.591·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(106.203\)
Root analytic conductor: \(10.3055\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8023436708\)
\(L(\frac12)\) \(\approx\) \(0.8023436708\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 34iT - 343T^{2} \)
11 \( 1 + 16T + 1.33e3T^{2} \)
13 \( 1 + 58iT - 2.19e3T^{2} \)
17 \( 1 - 70iT - 4.91e3T^{2} \)
19 \( 1 + 4T + 6.85e3T^{2} \)
23 \( 1 + 134iT - 1.21e4T^{2} \)
29 \( 1 + 242T + 2.43e4T^{2} \)
31 \( 1 - 100T + 2.97e4T^{2} \)
37 \( 1 + 438iT - 5.06e4T^{2} \)
41 \( 1 - 138T + 6.89e4T^{2} \)
43 \( 1 + 178iT - 7.95e4T^{2} \)
47 \( 1 + 22iT - 1.03e5T^{2} \)
53 \( 1 - 162iT - 1.48e5T^{2} \)
59 \( 1 + 268T + 2.05e5T^{2} \)
61 \( 1 - 250T + 2.26e5T^{2} \)
67 \( 1 - 422iT - 3.00e5T^{2} \)
71 \( 1 - 852T + 3.57e5T^{2} \)
73 \( 1 + 306iT - 3.89e5T^{2} \)
79 \( 1 - 456T + 4.93e5T^{2} \)
83 \( 1 - 434iT - 5.71e5T^{2} \)
89 \( 1 + 726T + 7.04e5T^{2} \)
97 \( 1 - 1.37e3iT - 9.12e5T^{2} \)
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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

−8.216653917690370161414310019503, −7.70824885489226415366625975491, −7.01702403633827120924261003847, −6.07891544147464450692316539687, −5.18062215942209140810155891597, −4.11095301543375202653994383317, −3.60834072190163192861077403549, −2.33241406423316551513256639311, −0.986169811466296005768752808220, −0.18587571312521825345759844268, 1.60949101075104289735804464423, 2.45697838985817334358055238736, 3.32107028601643397210609178351, 4.64639228900083435063532685245, 5.34619753767099391865582756977, 6.08439767770628909758181768335, 6.93885057789568231693330367550, 7.904493480950619413955930623874, 8.655594404750326920676172206107, 9.497600651252117366081922348247

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