Properties

Label 2-92400-1.1-c1-0-113
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s + 11-s − 5·13-s + 3·17-s − 7·19-s − 21-s + 4·23-s − 27-s − 4·29-s + 6·31-s − 33-s + 7·37-s + 5·39-s − 3·41-s − 8·43-s + 12·47-s + 49-s − 3·51-s − 7·53-s + 7·57-s + 7·61-s + 63-s − 9·67-s − 4·69-s + 7·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.38·13-s + 0.727·17-s − 1.60·19-s − 0.218·21-s + 0.834·23-s − 0.192·27-s − 0.742·29-s + 1.07·31-s − 0.174·33-s + 1.15·37-s + 0.800·39-s − 0.468·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.420·51-s − 0.961·53-s + 0.927·57-s + 0.896·61-s + 0.125·63-s − 1.09·67-s − 0.481·69-s + 0.830·71-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: \(1\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.20101528258025, −13.49208643671935, −13.00930752585093, −12.58453944488329, −12.01815651705983, −11.75843018453670, −11.12796065867626, −10.62987335055648, −10.20556622187462, −9.656062675721980, −9.213976335184547, −8.511238827414554, −8.057847805417824, −7.448636225473542, −6.983831250226052, −6.477786148897444, −5.852388974945418, −5.365732512094866, −4.634508273905891, −4.473389212234562, −3.681098471696418, −2.872763476882409, −2.301216180509691, −1.607763946614221, −0.8140280721213798, 0, 0.8140280721213798, 1.607763946614221, 2.301216180509691, 2.872763476882409, 3.681098471696418, 4.473389212234562, 4.634508273905891, 5.365732512094866, 5.852388974945418, 6.477786148897444, 6.983831250226052, 7.448636225473542, 8.057847805417824, 8.511238827414554, 9.213976335184547, 9.656062675721980, 10.20556622187462, 10.62987335055648, 11.12796065867626, 11.75843018453670, 12.01815651705983, 12.58453944488329, 13.00930752585093, 13.49208643671935, 14.20101528258025

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