Properties

Label 2-120e2-1.1-c1-0-5
Degree $2$
Conductor $14400$
Sign $1$
Analytic cond. $114.984$
Root an. cond. $10.7230$
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  − 4·7-s − 4·13-s + 8·19-s − 4·23-s − 6·29-s − 8·31-s + 4·37-s − 6·41-s − 4·43-s + 4·47-s + 9·49-s − 12·53-s + 6·61-s − 12·67-s + 16·71-s − 8·79-s − 12·83-s + 10·89-s + 16·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.10·13-s + 1.83·19-s − 0.834·23-s − 1.11·29-s − 1.43·31-s + 0.657·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s − 1.64·53-s + 0.768·61-s − 1.46·67-s + 1.89·71-s − 0.900·79-s − 1.31·83-s + 1.05·89-s + 1.67·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(114.984\)
Root analytic conductor: \(10.7230\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{14400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 14400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8092674744\)
\(L(\frac12)\) \(\approx\) \(0.8092674744\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−16.08269000886958, −15.68379251201532, −15.07447502699330, −14.31930069113060, −13.98098885814289, −13.08162202242730, −12.91316352939579, −12.18295545827683, −11.70189707621302, −11.07399568047675, −10.13369219763784, −9.818147068102598, −9.391561602214619, −8.806834648523250, −7.693712555721843, −7.444091017121557, −6.760634872885833, −6.042831217845396, −5.460496371088887, −4.814591396990432, −3.745738135564742, −3.346379311407920, −2.591217227405881, −1.671323366428086, −0.3811642083682741, 0.3811642083682741, 1.671323366428086, 2.591217227405881, 3.346379311407920, 3.745738135564742, 4.814591396990432, 5.460496371088887, 6.042831217845396, 6.760634872885833, 7.444091017121557, 7.693712555721843, 8.806834648523250, 9.391561602214619, 9.818147068102598, 10.13369219763784, 11.07399568047675, 11.70189707621302, 12.18295545827683, 12.91316352939579, 13.08162202242730, 13.98098885814289, 14.31930069113060, 15.07447502699330, 15.68379251201532, 16.08269000886958

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