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 − 2·17-s + 8·19-s − 4·23-s − 6·29-s − 4·31-s + 2·37-s + 2·41-s − 12·43-s + 49-s + 2·53-s − 4·59-s + 6·61-s − 4·67-s + 8·71-s − 6·73-s + 4·77-s + 16·79-s + 4·83-s + 18·89-s − 6·91-s − 6·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s − 1.11·29-s − 0.718·31-s + 0.328·37-s + 0.312·41-s − 1.82·43-s + 1/7·49-s + 0.274·53-s − 0.520·59-s + 0.768·61-s − 0.488·67-s + 0.949·71-s − 0.702·73-s + 0.455·77-s + 1.80·79-s + 0.439·83-s + 1.90·89-s − 0.628·91-s − 0.609·97-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

\(L(1)\) \(\approx\) \(1.701593998\)
\(L(\frac12)\) \(\approx\) \(1.701593998\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.37089931121090, −15.01941744952239, −14.13785244250490, −13.53614438852314, −13.38395897719698, −12.83365385178771, −12.08319315751276, −11.51995584616986, −11.00957345942141, −10.53227793946816, −9.867845515622848, −9.346895234950957, −8.790472843880461, −7.971381818001292, −7.782233865053584, −6.918458659835271, −6.353850639586943, −5.533819334707875, −5.382338980768763, −4.390451139370367, −3.535160791349165, −3.280976372045098, −2.282982389584438, −1.515853253627570, −0.5209745419639666, 0.5209745419639666, 1.515853253627570, 2.282982389584438, 3.280976372045098, 3.535160791349165, 4.390451139370367, 5.382338980768763, 5.533819334707875, 6.353850639586943, 6.918458659835271, 7.782233865053584, 7.971381818001292, 8.790472843880461, 9.346895234950957, 9.867845515622848, 10.53227793946816, 11.00957345942141, 11.51995584616986, 12.08319315751276, 12.83365385178771, 13.38395897719698, 13.53614438852314, 14.13785244250490, 15.01941744952239, 15.37089931121090

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