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Properties

Label 2-270480-1.1-c1-0-71
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 + 2·11-s − 4·13-s + 15-s + 6·17-s − 4·19-s + 23-s + 25-s + 27-s + 2·29-s + 2·31-s + 2·33-s + 4·37-s − 4·39-s + 2·41-s − 4·43-s + 45-s − 4·47-s + 6·51-s − 12·53-s + 2·55-s − 4·57-s + 6·59-s + 10·61-s − 4·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.348·33-s + 0.657·37-s − 0.640·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s − 0.583·47-s + 0.840·51-s − 1.64·53-s + 0.269·55-s − 0.529·57-s + 0.781·59-s + 1.28·61-s − 0.496·65-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

\(L(1)\) \(\approx\) \(3.988095201\)
\(L(\frac12)\) \(\approx\) \(3.988095201\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−12.80189337917635, −12.39141622573718, −11.98282101467615, −11.39604424197187, −10.96183983021359, −10.27589324600253, −9.900011911376424, −9.626901962057041, −9.197960061605332, −8.534260253252394, −8.069434096964122, −7.827612248063107, −7.034072294819578, −6.749432436325991, −6.225922778484092, −5.612915444464808, −5.064370074506061, −4.663110034223901, −3.989805983361778, −3.519035487681836, −2.874457161594470, −2.441794263508440, −1.825457622730642, −1.218613370682792, −0.5293373387729185, 0.5293373387729185, 1.218613370682792, 1.825457622730642, 2.441794263508440, 2.874457161594470, 3.519035487681836, 3.989805983361778, 4.663110034223901, 5.064370074506061, 5.612915444464808, 6.225922778484092, 6.749432436325991, 7.034072294819578, 7.827612248063107, 8.069434096964122, 8.534260253252394, 9.197960061605332, 9.626901962057041, 9.900011911376424, 10.27589324600253, 10.96183983021359, 11.39604424197187, 11.98282101467615, 12.39141622573718, 12.80189337917635

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