Properties

Label 2-1560-1560.389-c0-0-4
Degree $2$
Conductor $1560$
Sign $1$
Analytic cond. $0.778541$
Root an. cond. $0.882349$
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 + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s − 13-s + 15-s + 16-s − 18-s + 20-s − 24-s + 25-s + 26-s + 27-s − 30-s − 32-s + 36-s − 39-s − 40-s + 2·41-s − 2·43-s + 45-s + 48-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s − 13-s + 15-s + 16-s − 18-s + 20-s − 24-s + 25-s + 26-s + 27-s − 30-s − 32-s + 36-s − 39-s − 40-s + 2·41-s − 2·43-s + 45-s + 48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.778541\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1560} (389, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.198582535\)
\(L(\frac12)\) \(\approx\) \(1.198582535\)
\(L(1)\) 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 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 + T \)
good7 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )^{2} \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 + T )^{2} \)
73 \( 1 + T^{2} \)
79 \( ( 1 + T )^{2} \)
83 \( ( 1 - T )^{2} \)
89 \( ( 1 + T )^{2} \)
97 \( 1 + T^{2} \)
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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

−9.617491577200472130034388989993, −8.941424809775066615634410648681, −8.194792527080159563460027048944, −7.37310148028000106128810082902, −6.71614466108682143295487482295, −5.74993956813084240037296754846, −4.61456148074193800106382257507, −3.16313911805192908399185715721, −2.40660888618246790221497614286, −1.50135032922634249265141050625, 1.50135032922634249265141050625, 2.40660888618246790221497614286, 3.16313911805192908399185715721, 4.61456148074193800106382257507, 5.74993956813084240037296754846, 6.71614466108682143295487482295, 7.37310148028000106128810082902, 8.194792527080159563460027048944, 8.941424809775066615634410648681, 9.617491577200472130034388989993

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