Properties

Label 2-360-1.1-c1-0-2
Degree $2$
Conductor $360$
Sign $1$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
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  + 5-s + 2·7-s − 2·11-s + 4·13-s + 2·17-s + 4·19-s − 8·23-s + 25-s + 10·29-s + 4·31-s + 2·35-s − 8·43-s − 8·47-s − 3·49-s − 6·53-s − 2·55-s + 14·59-s − 14·61-s + 4·65-s − 4·67-s − 12·71-s + 6·73-s − 4·77-s − 12·79-s − 4·83-s + 2·85-s + 12·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s − 0.603·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.85·29-s + 0.718·31-s + 0.338·35-s − 1.21·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s − 0.269·55-s + 1.82·59-s − 1.79·61-s + 0.496·65-s − 0.488·67-s − 1.42·71-s + 0.702·73-s − 0.455·77-s − 1.35·79-s − 0.439·83-s + 0.216·85-s + 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.568979760\)
\(L(\frac12)\) \(\approx\) \(1.568979760\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−11.48759471734628960075235640364, −10.42977374464218618259946458906, −9.789609061778077574844038624674, −8.430531363871330384753929024064, −7.942341734598117376456164094334, −6.53876572552795316440603698045, −5.59184082567252664246215870943, −4.54321125337695258384863067085, −3.08000224242427569012154075457, −1.50557401701584832617156823174, 1.50557401701584832617156823174, 3.08000224242427569012154075457, 4.54321125337695258384863067085, 5.59184082567252664246215870943, 6.53876572552795316440603698045, 7.942341734598117376456164094334, 8.430531363871330384753929024064, 9.789609061778077574844038624674, 10.42977374464218618259946458906, 11.48759471734628960075235640364

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