Properties

Label 2-1536-3.2-c0-0-0
Degree $2$
Conductor $1536$
Sign $-1$
Analytic cond. $0.766563$
Root an. cond. $0.875536$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + 1.41i·5-s − 1.41·7-s − 9-s − 1.41·15-s − 1.41i·21-s − 1.00·25-s i·27-s + 1.41i·29-s − 1.41·31-s − 2.00i·35-s − 1.41i·45-s + 1.00·49-s − 1.41i·53-s + 2i·59-s + ⋯
L(s)  = 1  + i·3-s + 1.41i·5-s − 1.41·7-s − 9-s − 1.41·15-s − 1.41i·21-s − 1.00·25-s i·27-s + 1.41i·29-s − 1.41·31-s − 2.00i·35-s − 1.41i·45-s + 1.00·49-s − 1.41i·53-s + 2i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $-1$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6306205026\)
\(L(\frac12)\) \(\approx\) \(0.6306205026\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
good5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 2T + T^{2} \)
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

−10.11694378337472505584587902867, −9.413489807095613835543953171602, −8.737814088621360681779959284018, −7.46336340747380044247858058757, −6.73824302184038454308079115223, −6.06721125677170315531463711802, −5.15416656696291014008391082304, −3.77209691869147099429817644126, −3.33399801338009607358460732162, −2.48685273683211512943679477018, 0.47723753458551672793786135729, 1.83323515450409766242867810768, 3.05760073405233980040513067642, 4.14019518564257500800259096714, 5.32786036996171564093830852157, 6.01164636642245522598555766092, 6.79086485975441440948786605513, 7.70108622366752682736378142173, 8.444171678897054832532475526433, 9.248646551286118497724882773482

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