Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 4·11-s − 6·13-s + 2·17-s + 4·19-s − 8·23-s + 2·29-s + 10·37-s + 6·41-s − 4·43-s + 49-s + 6·53-s + 4·59-s + 6·61-s + 4·67-s + 8·71-s − 10·73-s + 4·77-s + 4·83-s + 6·89-s + 6·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 0.371·29-s + 1.64·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s + 0.520·59-s + 0.768·61-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.455·77-s + 0.439·83-s + 0.635·89-s + 0.628·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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: \(1\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.70659764964587, −15.10137009860046, −14.41530418705566, −14.22044460670655, −13.35978730707895, −12.97735713181105, −12.41054791449810, −11.82065827190064, −11.52926327964396, −10.41947962907158, −10.26476601082378, −9.624567245155726, −9.290878361187269, −8.190948290804764, −7.865032727282517, −7.399677190972346, −6.725433637565727, −5.903399551804274, −5.433020615831697, −4.832482907751441, −4.149036223814881, −3.326024620823437, −2.540560804006331, −2.220381897195115, −0.8879255454605991, 0, 0.8879255454605991, 2.220381897195115, 2.540560804006331, 3.326024620823437, 4.149036223814881, 4.832482907751441, 5.433020615831697, 5.903399551804274, 6.725433637565727, 7.399677190972346, 7.865032727282517, 8.190948290804764, 9.290878361187269, 9.624567245155726, 10.26476601082378, 10.41947962907158, 11.52926327964396, 11.82065827190064, 12.41054791449810, 12.97735713181105, 13.35978730707895, 14.22044460670655, 14.41530418705566, 15.10137009860046, 15.70659764964587

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