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Properties

Label 2-2160-15.14-c0-0-3
Degree $2$
Conductor $2160$
Sign $1$
Analytic cond. $1.07798$
Root an. cond. $1.03825$
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  + 5-s − 17-s + 19-s + 23-s + 25-s + 31-s − 2·47-s + 49-s − 53-s − 61-s + 79-s + 83-s − 85-s + 95-s − 2·107-s − 109-s + 2·113-s + 115-s + ⋯
L(s)  = 1  + 5-s − 17-s + 19-s + 23-s + 25-s + 31-s − 2·47-s + 49-s − 53-s − 61-s + 79-s + 83-s − 85-s + 95-s − 2·107-s − 109-s + 2·113-s + 115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2160} (1889, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.430607846\)
\(L(\frac12)\) \(\approx\) \(1.430607846\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.314755982477054221143082758028, −8.646261674320724763652826765045, −7.70301884847421054867270243950, −6.76350400436940707631919213332, −6.22901279508452714497602617344, −5.22337244520440055754597376928, −4.64098926891498131804169404996, −3.31321653895451706956557173084, −2.44418642050525191040624672666, −1.30675294302360740766815383931, 1.30675294302360740766815383931, 2.44418642050525191040624672666, 3.31321653895451706956557173084, 4.64098926891498131804169404996, 5.22337244520440055754597376928, 6.22901279508452714497602617344, 6.76350400436940707631919213332, 7.70301884847421054867270243950, 8.646261674320724763652826765045, 9.314755982477054221143082758028

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