Properties

Label 2-270-1.1-c3-0-12
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 5·5-s − 4·7-s − 8·8-s − 10·10-s − 42·11-s + 20·13-s + 8·14-s + 16·16-s − 93·17-s + 59·19-s + 20·20-s + 84·22-s − 9·23-s + 25·25-s − 40·26-s − 16·28-s − 120·29-s + 47·31-s − 32·32-s + 186·34-s − 20·35-s − 262·37-s − 118·38-s − 40·40-s − 126·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.215·7-s − 0.353·8-s − 0.316·10-s − 1.15·11-s + 0.426·13-s + 0.152·14-s + 1/4·16-s − 1.32·17-s + 0.712·19-s + 0.223·20-s + 0.814·22-s − 0.0815·23-s + 1/5·25-s − 0.301·26-s − 0.107·28-s − 0.768·29-s + 0.272·31-s − 0.176·32-s + 0.938·34-s − 0.0965·35-s − 1.16·37-s − 0.503·38-s − 0.158·40-s − 0.479·41-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: $\chi_{270} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 270,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 4 T + p^{3} T^{2} \)
11 \( 1 + 42 T + p^{3} T^{2} \)
13 \( 1 - 20 T + p^{3} T^{2} \)
17 \( 1 + 93 T + p^{3} T^{2} \)
19 \( 1 - 59 T + p^{3} T^{2} \)
23 \( 1 + 9 T + p^{3} T^{2} \)
29 \( 1 + 120 T + p^{3} T^{2} \)
31 \( 1 - 47 T + p^{3} T^{2} \)
37 \( 1 + 262 T + p^{3} T^{2} \)
41 \( 1 + 126 T + p^{3} T^{2} \)
43 \( 1 + 178 T + p^{3} T^{2} \)
47 \( 1 + 144 T + p^{3} T^{2} \)
53 \( 1 + 741 T + p^{3} T^{2} \)
59 \( 1 - 444 T + p^{3} T^{2} \)
61 \( 1 - 221 T + p^{3} T^{2} \)
67 \( 1 + 538 T + p^{3} T^{2} \)
71 \( 1 + 690 T + p^{3} T^{2} \)
73 \( 1 + 1126 T + p^{3} T^{2} \)
79 \( 1 - 665 T + p^{3} T^{2} \)
83 \( 1 + 75 T + p^{3} T^{2} \)
89 \( 1 - 1086 T + p^{3} T^{2} \)
97 \( 1 - 1544 T + p^{3} T^{2} \)
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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

−10.82387933032610466159230458133, −10.05188959390701941730898889448, −9.110335073730831573137237992067, −8.209429919550453124633346337166, −7.13752066718307239842628501035, −6.10908552591613562674787956181, −4.94045888314730274073260402257, −3.16781717532362284854674510984, −1.84338974082293675312061668138, 0, 1.84338974082293675312061668138, 3.16781717532362284854674510984, 4.94045888314730274073260402257, 6.10908552591613562674787956181, 7.13752066718307239842628501035, 8.209429919550453124633346337166, 9.110335073730831573137237992067, 10.05188959390701941730898889448, 10.82387933032610466159230458133

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