Properties

Label 2-1560-1.1-c1-0-10
Degree $2$
Conductor $1560$
Sign $1$
Analytic cond. $12.4566$
Root an. cond. $3.52939$
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  + 3-s + 5-s − 2·7-s + 9-s + 13-s + 15-s − 4·17-s + 6·19-s − 2·21-s + 6·23-s + 25-s + 27-s + 4·29-s + 8·31-s − 2·35-s − 6·37-s + 39-s + 6·41-s + 4·43-s + 45-s + 8·47-s − 3·49-s − 4·51-s + 2·53-s + 6·57-s − 2·61-s − 2·63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 0.970·17-s + 1.37·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 1.43·31-s − 0.338·35-s − 0.986·37-s + 0.160·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s + 0.794·57-s − 0.256·61-s − 0.251·63-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.211709898\)
\(L(\frac12)\) \(\approx\) \(2.211709898\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−9.253196033913166006878022808771, −8.911154955866250331585484293812, −7.85527228572738557890363868424, −6.97009190378829909782679476482, −6.34097146356439333020723004154, −5.31067242308855861340393122591, −4.34398567641935149508342927410, −3.21361498361474295842162005397, −2.54120816973867735274208270290, −1.07841227830594152864184443277, 1.07841227830594152864184443277, 2.54120816973867735274208270290, 3.21361498361474295842162005397, 4.34398567641935149508342927410, 5.31067242308855861340393122591, 6.34097146356439333020723004154, 6.97009190378829909782679476482, 7.85527228572738557890363868424, 8.911154955866250331585484293812, 9.253196033913166006878022808771

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