Properties

Label 2-92400-1.1-c1-0-101
Degree $2$
Conductor $92400$
Sign $1$
Analytic cond. $737.817$
Root an. cond. $27.1628$
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 − 7-s + 9-s + 11-s + 2·17-s + 6·19-s − 21-s + 8·23-s + 27-s − 4·31-s + 33-s + 2·37-s + 8·41-s + 8·43-s − 12·47-s + 49-s + 2·51-s + 10·53-s + 6·57-s + 12·59-s − 2·61-s − 63-s − 4·67-s + 8·69-s + 8·71-s + 12·73-s − 77-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.485·17-s + 1.37·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s − 0.718·31-s + 0.174·33-s + 0.328·37-s + 1.24·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s + 0.794·57-s + 1.56·59-s − 0.256·61-s − 0.125·63-s − 0.488·67-s + 0.963·69-s + 0.949·71-s + 1.40·73-s − 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(737.817\)
Root analytic conductor: \(27.1628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{92400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.318079143\)
\(L(\frac12)\) \(\approx\) \(4.318079143\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.86348619935002, −13.34768047321761, −12.83127600182937, −12.58038365809465, −11.83782014354276, −11.38594536391404, −10.92318665038399, −10.29995786959509, −9.701955743853750, −9.358991583820527, −8.976516635510695, −8.351401914773365, −7.712873917104618, −7.311731370492099, −6.871307197227875, −6.216556632643274, −5.544612026112233, −5.106519123025248, −4.459586882839817, −3.632709741560196, −3.394774122805925, −2.689631193862938, −2.115620863526685, −1.117723522739920, −0.7493824430760642, 0.7493824430760642, 1.117723522739920, 2.115620863526685, 2.689631193862938, 3.394774122805925, 3.632709741560196, 4.459586882839817, 5.106519123025248, 5.544612026112233, 6.216556632643274, 6.871307197227875, 7.311731370492099, 7.712873917104618, 8.351401914773365, 8.976516635510695, 9.358991583820527, 9.701955743853750, 10.29995786959509, 10.92318665038399, 11.38594536391404, 11.83782014354276, 12.58038365809465, 12.83127600182937, 13.34768047321761, 13.86348619935002

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