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Properties

Label 2-270480-1.1-c1-0-99
Degree $2$
Conductor $270480$
Sign $1$
Analytic cond. $2159.79$
Root an. cond. $46.4735$
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 − 5-s + 9-s + 4·11-s − 2·13-s − 15-s + 2·17-s − 4·19-s + 23-s + 25-s + 27-s + 2·29-s + 4·33-s + 6·37-s − 2·39-s + 6·41-s + 8·43-s − 45-s + 4·47-s + 2·51-s + 2·53-s − 4·55-s − 4·57-s + 12·59-s + 2·61-s + 2·65-s − 8·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.937·41-s + 1.21·43-s − 0.149·45-s + 0.583·47-s + 0.280·51-s + 0.274·53-s − 0.539·55-s − 0.529·57-s + 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270480\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(2159.79\)
Root analytic conductor: \(46.4735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 270480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.173662289\)
\(L(\frac12)\) \(\approx\) \(4.173662289\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
23 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−12.87049018956502, −12.20132150966755, −12.00156702357443, −11.43721275311275, −10.98916850177026, −10.39542311799226, −10.02950764929643, −9.432302421064610, −9.004620817304997, −8.719138923362995, −8.121472641188472, −7.587026816851887, −7.252074374051328, −6.767595518759601, −6.041908187467203, −5.892867445116342, −4.915796854596701, −4.497709755951099, −4.043835895889576, −3.606481774551464, −2.964375347135429, −2.386601744702928, −1.899287705910034, −1.007063741673353, −0.6239238119406341, 0.6239238119406341, 1.007063741673353, 1.899287705910034, 2.386601744702928, 2.964375347135429, 3.606481774551464, 4.043835895889576, 4.497709755951099, 4.915796854596701, 5.892867445116342, 6.041908187467203, 6.767595518759601, 7.252074374051328, 7.587026816851887, 8.121472641188472, 8.719138923362995, 9.004620817304997, 9.432302421064610, 10.02950764929643, 10.39542311799226, 10.98916850177026, 11.43721275311275, 12.00156702357443, 12.20132150966755, 12.87049018956502

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