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 − 4·11-s − 6·13-s + 4·17-s − 6·19-s + 6·29-s + 4·31-s − 8·37-s − 10·41-s − 2·43-s − 10·47-s + 49-s + 14·53-s − 4·59-s − 8·61-s + 6·67-s − 2·71-s + 10·73-s + 4·77-s − 16·79-s + 8·83-s − 2·89-s + 6·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 1.66·13-s + 0.970·17-s − 1.37·19-s + 1.11·29-s + 0.718·31-s − 1.31·37-s − 1.56·41-s − 0.304·43-s − 1.45·47-s + 1/7·49-s + 1.92·53-s − 0.520·59-s − 1.02·61-s + 0.733·67-s − 0.237·71-s + 1.17·73-s + 0.455·77-s − 1.80·79-s + 0.878·83-s − 0.211·89-s + 0.628·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\(L(1)\) \(\approx\) \(0.5982553931\)
\(L(\frac12)\) \(\approx\) \(0.5982553931\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−15.34795981242328, −14.83529423250748, −14.41125007136450, −13.63242129642355, −13.27058439627822, −12.58326953627904, −12.11387304187130, −11.87178967225225, −10.81419031327526, −10.28749024922945, −10.09364497764978, −9.476237393236092, −8.576815378005185, −8.189126390703473, −7.622340529216099, −6.851530525012060, −6.556123093004926, −5.553006133504844, −5.094574761416009, −4.604926891465254, −3.704143576202206, −2.879588813294562, −2.484157091048770, −1.583732961932619, −0.2918651600211315, 0.2918651600211315, 1.583732961932619, 2.484157091048770, 2.879588813294562, 3.704143576202206, 4.604926891465254, 5.094574761416009, 5.553006133504844, 6.556123093004926, 6.851530525012060, 7.622340529216099, 8.189126390703473, 8.576815378005185, 9.476237393236092, 10.09364497764978, 10.28749024922945, 10.81419031327526, 11.87178967225225, 12.11387304187130, 12.58326953627904, 13.27058439627822, 13.63242129642355, 14.41125007136450, 14.83529423250748, 15.34795981242328

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