Properties

Label 1-1560-1560.389-r1-0-0
Degree $1$
Conductor $1560$
Sign $1$
Analytic cond. $167.645$
Root an. cond. $167.645$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 11-s + 17-s + 19-s + 23-s + 29-s − 31-s − 37-s + 41-s + 43-s − 47-s + 49-s − 53-s − 59-s − 61-s − 67-s + 71-s + 73-s − 77-s + 79-s + 83-s + 89-s + 97-s + ⋯
L(s)  = 1  + 7-s − 11-s + 17-s + 19-s + 23-s + 29-s − 31-s − 37-s + 41-s + 43-s − 47-s + 49-s − 53-s − 59-s − 61-s − 67-s + 71-s + 73-s − 77-s + 79-s + 83-s + 89-s + 97-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(167.645\)
Root analytic conductor: \(167.645\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1560} (389, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1560,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.678586127\)
\(L(\frac12)\) \(\approx\) \(2.678586127\)
\(L(1)\) \(\approx\) \(1.272645822\)
\(L(1)\) \(\approx\) \(1.272645822\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−20.56494222053179448885057869694, −19.59284936695937273811928474682, −18.742801984884157368814261243069, −18.09464453606712872558899500501, −17.50630610704424279718105894641, −16.58346721117840886091580314412, −15.849655682680298359751258933312, −15.08046806491675659149204126875, −14.27304979337813339210666427266, −13.720656860950465307743859403270, −12.67619570062849114179071037410, −12.06253107453793570366081579172, −11.06602025964258717290869495543, −10.60030168800036189359233384306, −9.59873748338549394108967876124, −8.77067197774125362291393296653, −7.723523662352727843363013374882, −7.52591712706016223144439496153, −6.20540153991903214303817441961, −5.18208357224033577072698876383, −4.873434727256469188523245721137, −3.5449664720224192613210780956, −2.733739807912684513598254555156, −1.63585249527135430180419793473, −0.7275425887162805901150346916, 0.7275425887162805901150346916, 1.63585249527135430180419793473, 2.733739807912684513598254555156, 3.5449664720224192613210780956, 4.873434727256469188523245721137, 5.18208357224033577072698876383, 6.20540153991903214303817441961, 7.52591712706016223144439496153, 7.723523662352727843363013374882, 8.77067197774125362291393296653, 9.59873748338549394108967876124, 10.60030168800036189359233384306, 11.06602025964258717290869495543, 12.06253107453793570366081579172, 12.67619570062849114179071037410, 13.720656860950465307743859403270, 14.27304979337813339210666427266, 15.08046806491675659149204126875, 15.849655682680298359751258933312, 16.58346721117840886091580314412, 17.50630610704424279718105894641, 18.09464453606712872558899500501, 18.742801984884157368814261243069, 19.59284936695937273811928474682, 20.56494222053179448885057869694

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