Properties

Label 2-270-1.1-c3-0-7
Degree $2$
Conductor $270$
Sign $1$
Analytic cond. $15.9305$
Root an. cond. $3.99130$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s − 5·5-s + 14·7-s + 8·8-s − 10·10-s − 3·11-s + 47·13-s + 28·14-s + 16·16-s + 39·17-s + 32·19-s − 20·20-s − 6·22-s + 99·23-s + 25·25-s + 94·26-s + 56·28-s − 51·29-s + 83·31-s + 32·32-s + 78·34-s − 70·35-s + 314·37-s + 64·38-s − 40·40-s + 108·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s − 0.0822·11-s + 1.00·13-s + 0.534·14-s + 1/4·16-s + 0.556·17-s + 0.386·19-s − 0.223·20-s − 0.0581·22-s + 0.897·23-s + 1/5·25-s + 0.709·26-s + 0.377·28-s − 0.326·29-s + 0.480·31-s + 0.176·32-s + 0.393·34-s − 0.338·35-s + 1.39·37-s + 0.273·38-s − 0.158·40-s + 0.411·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(15.9305\)
Root analytic conductor: \(3.99130\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.014685092\)
\(L(\frac12)\) \(\approx\) \(3.014685092\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 \)
5 \( 1 + p T \)
good7 \( 1 - 2 p T + p^{3} T^{2} \)
11 \( 1 + 3 T + p^{3} T^{2} \)
13 \( 1 - 47 T + p^{3} T^{2} \)
17 \( 1 - 39 T + p^{3} T^{2} \)
19 \( 1 - 32 T + p^{3} T^{2} \)
23 \( 1 - 99 T + p^{3} T^{2} \)
29 \( 1 + 51 T + p^{3} T^{2} \)
31 \( 1 - 83 T + p^{3} T^{2} \)
37 \( 1 - 314 T + p^{3} T^{2} \)
41 \( 1 - 108 T + p^{3} T^{2} \)
43 \( 1 - 299 T + p^{3} T^{2} \)
47 \( 1 + 531 T + p^{3} T^{2} \)
53 \( 1 + 564 T + p^{3} T^{2} \)
59 \( 1 + 12 T + p^{3} T^{2} \)
61 \( 1 - 230 T + p^{3} T^{2} \)
67 \( 1 + 4 p T + p^{3} T^{2} \)
71 \( 1 + 120 T + p^{3} T^{2} \)
73 \( 1 - 1106 T + p^{3} T^{2} \)
79 \( 1 + 739 T + p^{3} T^{2} \)
83 \( 1 + 1086 T + p^{3} T^{2} \)
89 \( 1 - 120 T + p^{3} T^{2} \)
97 \( 1 + 1642 T + p^{3} T^{2} \)
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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

−11.34106330990542563348986185600, −11.03681396601531768652904083476, −9.641447202588911584911439410560, −8.346704299076133895999256290881, −7.58292040328467036865502877057, −6.35553036251532870381636664131, −5.23895334946413519346716508743, −4.20292216471389383059227878191, −3.00896933479992684974418391341, −1.27242072369560111817595953244, 1.27242072369560111817595953244, 3.00896933479992684974418391341, 4.20292216471389383059227878191, 5.23895334946413519346716508743, 6.35553036251532870381636664131, 7.58292040328467036865502877057, 8.346704299076133895999256290881, 9.641447202588911584911439410560, 11.03681396601531768652904083476, 11.34106330990542563348986185600

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