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 − 6·11-s − 2·13-s − 2·17-s − 4·19-s + 4·23-s + 2·29-s + 2·31-s + 10·37-s + 6·41-s − 2·43-s − 2·47-s + 49-s + 6·53-s + 4·59-s − 12·61-s + 10·67-s − 12·71-s + 2·73-s + 6·77-s + 16·79-s + 12·83-s + 14·89-s + 2·91-s − 18·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.80·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 0.371·29-s + 0.359·31-s + 1.64·37-s + 0.937·41-s − 0.304·43-s − 0.291·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s − 1.53·61-s + 1.22·67-s − 1.42·71-s + 0.234·73-s + 0.683·77-s + 1.80·79-s + 1.31·83-s + 1.48·89-s + 0.209·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.51348087489482, −15.10188882441420, −14.78676592501211, −13.89001842926704, −13.35268484084209, −12.99394791751469, −12.53933260778492, −11.94912315666613, −11.04040167696378, −10.85347514621506, −10.16885659658088, −9.714392708232562, −9.014176085808490, −8.444329752999385, −7.748743518379127, −7.449115064816102, −6.556801143327780, −6.134502540202862, −5.302143615660665, −4.816223595792053, −4.225961117983937, −3.270406993392790, −2.544229687413557, −2.246668442901821, −0.8730980542688705, 0, 0.8730980542688705, 2.246668442901821, 2.544229687413557, 3.270406993392790, 4.225961117983937, 4.816223595792053, 5.302143615660665, 6.134502540202862, 6.556801143327780, 7.449115064816102, 7.748743518379127, 8.444329752999385, 9.014176085808490, 9.714392708232562, 10.16885659658088, 10.85347514621506, 11.04040167696378, 11.94912315666613, 12.53933260778492, 12.99394791751469, 13.35268484084209, 13.89001842926704, 14.78676592501211, 15.10188882441420, 15.51348087489482

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