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Properties

Label 2-360-40.19-c0-0-0
Degree $2$
Conductor $360$
Sign $1$
Analytic cond. $0.179663$
Root an. cond. $0.423867$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s + 16-s − 2·19-s + 20-s + 2·23-s + 25-s − 32-s + 2·38-s − 40-s − 2·46-s − 2·47-s − 49-s − 50-s − 2·53-s + 64-s − 2·76-s + 80-s + 2·92-s + 2·94-s − 2·95-s + 98-s + 100-s + 2·106-s + ⋯
L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s + 16-s − 2·19-s + 20-s + 2·23-s + 25-s − 32-s + 2·38-s − 40-s − 2·46-s − 2·47-s − 49-s − 50-s − 2·53-s + 64-s − 2·76-s + 80-s + 2·92-s + 2·94-s − 2·95-s + 98-s + 100-s + 2·106-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.179663\)
Root analytic conductor: \(0.423867\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{360} (19, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6223734010\)
\(L(\frac12)\) \(\approx\) \(0.6223734010\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 + T )^{2} \)
23 \( ( 1 - T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 + T )^{2} \)
53 \( ( 1 + T )^{2} \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−11.23286487780615393400666536909, −10.68187749341618013571341686120, −9.736715305155205799149249418507, −8.985628989755838456102110920539, −8.168689006898785089475640065234, −6.83220418461643527692503578463, −6.24191225269475381330774949119, −4.92433240212012765692717661545, −2.98885868798805186176990190388, −1.72222281807707128570367328671, 1.72222281807707128570367328671, 2.98885868798805186176990190388, 4.92433240212012765692717661545, 6.24191225269475381330774949119, 6.83220418461643527692503578463, 8.168689006898785089475640065234, 8.985628989755838456102110920539, 9.736715305155205799149249418507, 10.68187749341618013571341686120, 11.23286487780615393400666536909

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