Properties

Label 2-62400-1.1-c1-0-198
Degree $2$
Conductor $62400$
Sign $1$
Analytic cond. $498.266$
Root an. cond. $22.3218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s − 4·11-s + 13-s − 6·17-s + 4·21-s + 4·23-s − 27-s + 6·29-s − 8·31-s + 4·33-s − 2·37-s − 39-s + 10·41-s − 4·43-s − 8·47-s + 9·49-s + 6·51-s − 2·53-s − 4·59-s − 14·61-s − 4·63-s − 12·67-s − 4·69-s − 8·71-s + 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 1.45·17-s + 0.872·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s − 0.328·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.840·51-s − 0.274·53-s − 0.520·59-s − 1.79·61-s − 0.503·63-s − 1.46·67-s − 0.481·69-s − 0.949·71-s + 1.17·73-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: $1$
Analytic conductor: \(498.266\)
Root analytic conductor: \(22.3218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{62400} (1, \cdot )$
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
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

−14.89525400683498, −14.22320179328799, −13.40577355290417, −13.22216622146172, −12.89470211392599, −12.33376247970666, −11.82183181415631, −11.00337290068401, −10.73820793415425, −10.38144719294556, −9.619264762196638, −9.201626119112376, −8.786639563315111, −7.958546737946582, −7.433724926725977, −6.789176126486129, −6.427930021854941, −5.929097382637909, −5.271335751926727, −4.700437029312747, −4.125240763369794, −3.256927308669433, −2.887227358816682, −2.161682445267374, −1.209592659864813, 0, 0, 1.209592659864813, 2.161682445267374, 2.887227358816682, 3.256927308669433, 4.125240763369794, 4.700437029312747, 5.271335751926727, 5.929097382637909, 6.427930021854941, 6.789176126486129, 7.433724926725977, 7.958546737946582, 8.786639563315111, 9.201626119112376, 9.619264762196638, 10.38144719294556, 10.73820793415425, 11.00337290068401, 11.82183181415631, 12.33376247970666, 12.89470211392599, 13.22216622146172, 13.40577355290417, 14.22320179328799, 14.89525400683498

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