Properties

Label 2-1800-1.1-c3-0-50
Degree $2$
Conductor $1800$
Sign $-1$
Analytic cond. $106.203$
Root an. cond. $10.3055$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.96·7-s − 44.6·11-s − 68.4·13-s + 90.1·17-s + 75.3·19-s − 11.0·23-s + 145.·29-s + 202.·31-s + 124.·37-s − 150.·41-s + 134.·43-s − 510.·47-s − 279.·49-s − 16.2·53-s − 399.·59-s − 486.·61-s − 353.·67-s + 76.9·71-s + 419.·73-s − 355.·77-s + 110.·79-s + 45.6·83-s − 724.·89-s − 545.·91-s − 539.·97-s − 1.48e3·101-s − 1.20e3·103-s + ⋯
L(s)  = 1  + 0.430·7-s − 1.22·11-s − 1.45·13-s + 1.28·17-s + 0.910·19-s − 0.100·23-s + 0.934·29-s + 1.17·31-s + 0.555·37-s − 0.575·41-s + 0.476·43-s − 1.58·47-s − 0.814·49-s − 0.0422·53-s − 0.880·59-s − 1.02·61-s − 0.644·67-s + 0.128·71-s + 0.673·73-s − 0.526·77-s + 0.157·79-s + 0.0603·83-s − 0.863·89-s − 0.627·91-s − 0.564·97-s − 1.46·101-s − 1.15·103-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(106.203\)
Root analytic conductor: \(10.3055\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1800,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 7.96T + 343T^{2} \)
11 \( 1 + 44.6T + 1.33e3T^{2} \)
13 \( 1 + 68.4T + 2.19e3T^{2} \)
17 \( 1 - 90.1T + 4.91e3T^{2} \)
19 \( 1 - 75.3T + 6.85e3T^{2} \)
23 \( 1 + 11.0T + 1.21e4T^{2} \)
29 \( 1 - 145.T + 2.43e4T^{2} \)
31 \( 1 - 202.T + 2.97e4T^{2} \)
37 \( 1 - 124.T + 5.06e4T^{2} \)
41 \( 1 + 150.T + 6.89e4T^{2} \)
43 \( 1 - 134.T + 7.95e4T^{2} \)
47 \( 1 + 510.T + 1.03e5T^{2} \)
53 \( 1 + 16.2T + 1.48e5T^{2} \)
59 \( 1 + 399.T + 2.05e5T^{2} \)
61 \( 1 + 486.T + 2.26e5T^{2} \)
67 \( 1 + 353.T + 3.00e5T^{2} \)
71 \( 1 - 76.9T + 3.57e5T^{2} \)
73 \( 1 - 419.T + 3.89e5T^{2} \)
79 \( 1 - 110.T + 4.93e5T^{2} \)
83 \( 1 - 45.6T + 5.71e5T^{2} \)
89 \( 1 + 724.T + 7.04e5T^{2} \)
97 \( 1 + 539.T + 9.12e5T^{2} \)
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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

−8.209260213831625359833940412969, −7.85837222637497821093908885621, −7.11947115161827970192025549215, −6.01459569840045718459781688105, −5.07056698608370365096314829609, −4.70497037221906598010057614924, −3.20700165223606679123208956723, −2.55787839678681633492736494908, −1.26424156442316019485929903653, 0, 1.26424156442316019485929903653, 2.55787839678681633492736494908, 3.20700165223606679123208956723, 4.70497037221906598010057614924, 5.07056698608370365096314829609, 6.01459569840045718459781688105, 7.11947115161827970192025549215, 7.85837222637497821093908885621, 8.209260213831625359833940412969

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