Properties

Label 2-62400-1.1-c1-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s + 4·11-s + 13-s − 2·17-s + 8·19-s + 4·21-s − 27-s − 6·29-s − 4·31-s − 4·33-s − 2·37-s − 39-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s + 2·51-s − 10·53-s − 8·57-s − 4·59-s + 2·61-s − 4·63-s − 16·67-s − 8·71-s − 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.485·17-s + 1.83·19-s + 0.872·21-s − 0.192·27-s − 1.11·29-s − 0.718·31-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 + 0.280·51-s − 1.37·53-s − 1.05·57-s − 0.520·59-s + 0.256·61-s − 0.503·63-s − 1.95·67-s − 0.949·71-s − 0.234·73-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 62400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9173362253\)
\(L(\frac12)\) \(\approx\) \(0.9173362253\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.16107788764010, −13.64925029106551, −13.24377135407827, −12.79962672404359, −12.07041053327919, −11.89863895327777, −11.28608460553366, −10.76269459676330, −10.11710197410330, −9.603239753904506, −9.224014047954817, −8.923602017513256, −7.926903507613855, −7.345747239349982, −6.855962767345110, −6.383866382528157, −5.952253267382424, −5.339977394325979, −4.702104913499659, −3.910378181355136, −3.366109793485534, −3.089498074718448, −1.875428993749397, −1.305212843070448, −0.3513693356622415, 0.3513693356622415, 1.305212843070448, 1.875428993749397, 3.089498074718448, 3.366109793485534, 3.910378181355136, 4.702104913499659, 5.339977394325979, 5.952253267382424, 6.383866382528157, 6.855962767345110, 7.345747239349982, 7.926903507613855, 8.923602017513256, 9.224014047954817, 9.603239753904506, 10.11710197410330, 10.76269459676330, 11.28608460553366, 11.89863895327777, 12.07041053327919, 12.79962672404359, 13.24377135407827, 13.64925029106551, 14.16107788764010

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