Properties

Label 2-1560-1560.389-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(0.6709971725\)
\(L(\frac12)\) \(\approx\) \(0.6709971725\)
\(L(1)\) 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

−9.687436797388697297566263881654, −9.014314449018880032663446816979, −8.122087241435947886900642964907, −7.09853387789511821313589912679, −6.38182309473371025789185944231, −5.85194413889033642024200046525, −4.95344269380575781900265179921, −3.52041895484861317305858587807, −2.11733092297523868146437438628, −1.12004538889761268642888049684, 1.12004538889761268642888049684, 2.11733092297523868146437438628, 3.52041895484861317305858587807, 4.95344269380575781900265179921, 5.85194413889033642024200046525, 6.38182309473371025789185944231, 7.09853387789511821313589912679, 8.122087241435947886900642964907, 9.014314449018880032663446816979, 9.687436797388697297566263881654

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