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Properties

Label 2-1800-8.3-c0-0-3
Degree $2$
Conductor $1800$
Sign $1$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 11-s + 16-s − 17-s − 19-s + 22-s + 32-s − 34-s − 38-s + 41-s − 2·43-s + 44-s + 49-s − 2·59-s + 64-s + 67-s − 68-s + 73-s − 76-s + 82-s − 83-s − 2·86-s + 88-s + 89-s − 2·97-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s + 11-s + 16-s − 17-s − 19-s + 22-s + 32-s − 34-s − 38-s + 41-s − 2·43-s + 44-s + 49-s − 2·59-s + 64-s + 67-s − 68-s + 73-s − 76-s + 82-s − 83-s − 2·86-s + 88-s + 89-s − 2·97-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1800} (451, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.202820647\)
\(L(\frac12)\) \(\approx\) \(2.202820647\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.465625098131355918188403281033, −8.642810312287221008585754063677, −7.75047902300715182823950980305, −6.67833116003745239717115701629, −6.42242258273585235599389438989, −5.33686562578059969849304683945, −4.40719362115053941726705538025, −3.81255898184309720005267252915, −2.65353599889505161743456569945, −1.62572052382060186085981353731, 1.62572052382060186085981353731, 2.65353599889505161743456569945, 3.81255898184309720005267252915, 4.40719362115053941726705538025, 5.33686562578059969849304683945, 6.42242258273585235599389438989, 6.67833116003745239717115701629, 7.75047902300715182823950980305, 8.642810312287221008585754063677, 9.465625098131355918188403281033

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