Properties

Label 2-360-1.1-c1-0-3
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 + 4·7-s − 6·13-s + 2·17-s + 4·19-s + 8·23-s + 25-s + 6·29-s + 4·35-s − 6·37-s − 10·41-s − 4·43-s − 8·47-s + 9·49-s − 10·53-s + 6·61-s − 6·65-s − 4·67-s − 14·73-s + 16·79-s − 12·83-s + 2·85-s − 2·89-s − 24·91-s + 4·95-s + 2·97-s + 14·101-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 1.66·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.676·35-s − 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s + 0.768·61-s − 0.744·65-s − 0.488·67-s − 1.63·73-s + 1.80·79-s − 1.31·83-s + 0.216·85-s − 0.211·89-s − 2.51·91-s + 0.410·95-s + 0.203·97-s + 1.39·101-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

\(L(1)\) \(\approx\) \(1.614650770\)
\(L(\frac12)\) \(\approx\) \(1.614650770\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−11.55011761506285336811259028906, −10.48496722029080345949520999093, −9.686400747849369995880150049093, −8.609245026512276947697146238421, −7.67822247307498044046677545125, −6.82423053307536595497404589251, −5.12597312940943561354086695880, −4.92167313563440441781925895939, −2.97865181251010540219662645668, −1.55129913823425505871935815725, 1.55129913823425505871935815725, 2.97865181251010540219662645668, 4.92167313563440441781925895939, 5.12597312940943561354086695880, 6.82423053307536595497404589251, 7.67822247307498044046677545125, 8.609245026512276947697146238421, 9.686400747849369995880150049093, 10.48496722029080345949520999093, 11.55011761506285336811259028906

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