↑

Properties

Label 2-504-8.5-c1-0-6
Degree $2$
Conductor $504$
Sign $-i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 0.707i)2-s + (0.999 − 1.73i)4-s + 1.41i·5-s + 7-s + 2.82i·8-s + (−1.00 − 1.73i)10-s + 1.41i·11-s + (−1.22 + 0.707i)14-s + (−2.00 − 3.46i)16-s + 2.44·17-s + 3.46i·19-s + (2.44 + 1.41i)20-s + (−1.00 − 1.73i)22-s − 2.44·23-s + 2.99·25-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (0.499 − 0.866i)4-s + 0.632i·5-s + 0.377·7-s + 0.999i·8-s + (−0.316 − 0.547i)10-s + 0.426i·11-s + (−0.327 + 0.188i)14-s + (−0.500 − 0.866i)16-s + 0.594·17-s + 0.794i·19-s + (0.547 + 0.316i)20-s + (−0.213 − 0.369i)22-s − 0.510·23-s + 0.599·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663149 + 0.663149i\)
\(L(\frac12)\) \(\approx\) \(0.663149 + 0.663149i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 1.41iT - 5T^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 2.44T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 2.44T + 23T^{2} \)
29 \( 1 - 5.65iT - 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 3.46iT - 37T^{2} \)
41 \( 1 - 7.34T + 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + 7.34T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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

−10.88647803107559495083050564201, −10.17847012695473798468895283702, −9.419251681092523110368778433845, −8.306111710647305147799432597359, −7.62395382306754769871625524029, −6.71650888863171536925222881926, −5.83002641205238675019185268288, −4.67390593224415943310808325140, −2.98741242597082494602617776280, −1.52482990367394484839681975845, 0.806470209863134182034020250678, 2.29771094923668618910981038416, 3.66858830125814669163406463903, 4.88488549499706473560029556004, 6.18657952027317844392775494797, 7.37900434091854391118146853593, 8.214664249397158635990184320013, 8.923718765542202228705175930740, 9.756945034786607727357287758069, 10.67648342791375220590733144635

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