Properties

Label 2-62400-1.1-c1-0-108
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 − 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: \(0\)
Selberg data: \((2,\ 62400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.058413778\)
\(L(\frac12)\) \(\approx\) \(5.058413778\)
\(L(\frac{3}{2})\) 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

−14.13402410791269, −13.87876909513631, −13.60826026942288, −12.63683832953987, −12.25042816940011, −11.70819890165527, −11.24483091567793, −10.77091952900483, −10.29636774206395, −9.427035232404143, −9.129134226537177, −8.558916461625108, −8.085633873554289, −7.747464241372526, −6.900548124082038, −6.487567284651897, −5.927037077249745, −5.068747008273989, −4.404526103867179, −4.259366927326495, −3.539084332714494, −2.478201415502130, −2.204054059401273, −1.346198393034542, −0.8010794985750460, 0.8010794985750460, 1.346198393034542, 2.204054059401273, 2.478201415502130, 3.539084332714494, 4.259366927326495, 4.404526103867179, 5.068747008273989, 5.927037077249745, 6.487567284651897, 6.900548124082038, 7.747464241372526, 8.085633873554289, 8.558916461625108, 9.129134226537177, 9.427035232404143, 10.29636774206395, 10.77091952900483, 11.24483091567793, 11.70819890165527, 12.25042816940011, 12.63683832953987, 13.60826026942288, 13.87876909513631, 14.13402410791269

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