Properties

Degree $2$
Conductor $62400$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 4·11-s + 13-s + 6·17-s − 4·19-s − 8·23-s − 27-s − 6·29-s − 8·31-s + 4·33-s − 10·37-s − 39-s − 6·41-s + 4·43-s − 7·49-s − 6·51-s − 10·53-s + 4·57-s − 4·59-s + 2·61-s − 12·67-s + 8·69-s + 16·71-s − 2·73-s − 16·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.45·17-s − 0.917·19-s − 1.66·23-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s − 1.64·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s − 49-s − 0.840·51-s − 1.37·53-s + 0.529·57-s − 0.520·59-s + 0.256·61-s − 1.46·67-s + 0.963·69-s + 1.89·71-s − 0.234·73-s − 1.80·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62400\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{62400} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 62400,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.63878251492721, −14.33227558823778, −13.79290401782970, −13.00700346668250, −12.84535217171812, −12.24675465462160, −11.82731779617667, −11.11816891019666, −10.75002519030502, −10.17716336099288, −9.915893235339292, −9.172661379763343, −8.536082846637560, −7.918374140365309, −7.608750819728159, −7.009916830253601, −6.190470204037298, −5.853390000166195, −5.249805316160757, −4.880854185749742, −3.893106195667164, −3.594927454688478, −2.770008960744422, −1.873612893606683, −1.486051582636199, 0, 0, 1.486051582636199, 1.873612893606683, 2.770008960744422, 3.594927454688478, 3.893106195667164, 4.880854185749742, 5.249805316160757, 5.853390000166195, 6.190470204037298, 7.009916830253601, 7.608750819728159, 7.918374140365309, 8.536082846637560, 9.172661379763343, 9.915893235339292, 10.17716336099288, 10.75002519030502, 11.11816891019666, 11.82731779617667, 12.24675465462160, 12.84535217171812, 13.00700346668250, 13.79290401782970, 14.33227558823778, 14.63878251492721

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