Properties

Label 2-1568-1.1-c1-0-14
Degree $2$
Conductor $1568$
Sign $1$
Analytic cond. $12.5205$
Root an. cond. $3.53843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 3·9-s − 4·13-s + 8·17-s + 11·25-s + 10·29-s + 2·37-s + 8·41-s − 12·45-s + 14·53-s − 12·61-s − 16·65-s − 16·73-s + 9·81-s + 32·85-s − 16·89-s − 8·97-s + 20·101-s − 6·109-s − 14·113-s + 12·117-s + ⋯
L(s)  = 1  + 1.78·5-s − 9-s − 1.10·13-s + 1.94·17-s + 11/5·25-s + 1.85·29-s + 0.328·37-s + 1.24·41-s − 1.78·45-s + 1.92·53-s − 1.53·61-s − 1.98·65-s − 1.87·73-s + 81-s + 3.47·85-s − 1.69·89-s − 0.812·97-s + 1.99·101-s − 0.574·109-s − 1.31·113-s + 1.10·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.279726292\)
\(L(\frac12)\) \(\approx\) \(2.279726292\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.548865919223462988821068539485, −8.795935640120407431962303331374, −7.88132253194711275835728190267, −6.91533496238080223154082214854, −5.88267197317121816107420623197, −5.59269710161924499254999286275, −4.66389023955673654317656217449, −3.01558964211138052679351660845, −2.46224298486979636156274494161, −1.13590008709403894614989037078, 1.13590008709403894614989037078, 2.46224298486979636156274494161, 3.01558964211138052679351660845, 4.66389023955673654317656217449, 5.59269710161924499254999286275, 5.88267197317121816107420623197, 6.91533496238080223154082214854, 7.88132253194711275835728190267, 8.795935640120407431962303331374, 9.548865919223462988821068539485

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