Properties

Degree $2$
Conductor $25200$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 6·11-s + 2·13-s + 4·17-s + 6·19-s + 2·29-s + 10·31-s + 4·37-s − 2·41-s − 4·43-s + 49-s + 6·53-s − 8·59-s − 2·61-s − 16·67-s + 10·71-s + 6·73-s + 6·77-s − 4·79-s − 8·83-s − 6·89-s − 2·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.80·11-s + 0.554·13-s + 0.970·17-s + 1.37·19-s + 0.371·29-s + 1.79·31-s + 0.657·37-s − 0.312·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s − 1.04·59-s − 0.256·61-s − 1.95·67-s + 1.18·71-s + 0.702·73-s + 0.683·77-s − 0.450·79-s − 0.878·83-s − 0.635·89-s − 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{25200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.945010387\)
\(L(\frac12)\) \(\approx\) \(1.945010387\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.42672448968787, −15.01803881081257, −14.06642295724661, −13.73005502909044, −13.31089456154457, −12.70421189879640, −12.11652010862228, −11.66681590936476, −10.94801953714280, −10.36134430209200, −9.944831999615322, −9.527658104132175, −8.559385869356506, −8.177653534057751, −7.565661604079648, −7.135428603333742, −6.154383177301171, −5.803235700831152, −5.054825875842787, −4.623590031076831, −3.563078300429563, −3.005776589837587, −2.542974745241943, −1.379939466878990, −0.5829631726033002, 0.5829631726033002, 1.379939466878990, 2.542974745241943, 3.005776589837587, 3.563078300429563, 4.623590031076831, 5.054825875842787, 5.803235700831152, 6.154383177301171, 7.135428603333742, 7.565661604079648, 8.177653534057751, 8.559385869356506, 9.527658104132175, 9.944831999615322, 10.36134430209200, 10.94801953714280, 11.66681590936476, 12.11652010862228, 12.70421189879640, 13.31089456154457, 13.73005502909044, 14.06642295724661, 15.01803881081257, 15.42672448968787

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