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Properties

Label 2-1800-15.14-c2-0-17
Degree $2$
Conductor $1800$
Sign $0.988 - 0.151i$
Analytic cond. $49.0464$
Root an. cond. $7.00331$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.16i·7-s − 3.05i·11-s − 5.16i·13-s + 17.8·17-s − 10.9·19-s − 10.5·23-s + 42.6i·29-s + 33.6·31-s − 3.48i·37-s − 34.3i·41-s − 80.2i·43-s + 0.458·47-s + 39·49-s + 12.3·53-s + 12.5i·59-s + ⋯
L(s)  = 1  + 0.451i·7-s − 0.277i·11-s − 0.397i·13-s + 1.05·17-s − 0.577·19-s − 0.460·23-s + 1.46i·29-s + 1.08·31-s − 0.0942i·37-s − 0.838i·41-s − 1.86i·43-s + 0.00976·47-s + 0.795·49-s + 0.232·53-s + 0.212i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.988 - 0.151i$
Analytic conductor: \(49.0464\)
Root analytic conductor: \(7.00331\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1),\ 0.988 - 0.151i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.969924191\)
\(L(\frac12)\) \(\approx\) \(1.969924191\)
\(L(2)\) 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 \( 1 - 3.16iT - 49T^{2} \)
11 \( 1 + 3.05iT - 121T^{2} \)
13 \( 1 + 5.16iT - 169T^{2} \)
17 \( 1 - 17.8T + 289T^{2} \)
19 \( 1 + 10.9T + 361T^{2} \)
23 \( 1 + 10.5T + 529T^{2} \)
29 \( 1 - 42.6iT - 841T^{2} \)
31 \( 1 - 33.6T + 961T^{2} \)
37 \( 1 + 3.48iT - 1.36e3T^{2} \)
41 \( 1 + 34.3iT - 1.68e3T^{2} \)
43 \( 1 + 80.2iT - 1.84e3T^{2} \)
47 \( 1 - 0.458T + 2.20e3T^{2} \)
53 \( 1 - 12.3T + 2.80e3T^{2} \)
59 \( 1 - 12.5iT - 3.48e3T^{2} \)
61 \( 1 + 6.27T + 3.72e3T^{2} \)
67 \( 1 - 71.8iT - 4.48e3T^{2} \)
71 \( 1 - 77.7iT - 5.04e3T^{2} \)
73 \( 1 - 31.5iT - 5.32e3T^{2} \)
79 \( 1 - 125.T + 6.24e3T^{2} \)
83 \( 1 - 74.3T + 6.88e3T^{2} \)
89 \( 1 - 68.8iT - 7.92e3T^{2} \)
97 \( 1 + 39.5iT - 9.40e3T^{2} \)
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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

−8.923325319638195532779456172678, −8.463678704331757943344883295896, −7.55382319067653295806471544262, −6.74625580745607104453269444390, −5.75139779011304043422991828581, −5.23503578151297600553149051714, −4.04025248408083977774625814217, −3.15517604390094506165939582480, −2.11982772484812766355984373514, −0.794082839927001764021142147088, 0.75251714555345412733772779599, 1.99078335255926580143961474160, 3.12838014737432291963548321442, 4.16790273549520093701422078549, 4.83237260899316403954790280972, 6.03628062405152725189703491908, 6.56013432764584939926505612063, 7.73329450772714036111224415758, 8.029946554848822984306443166059, 9.190174788975631707412489646846

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