↑

Properties

Label 2-504-24.11-c1-0-6
Degree $2$
Conductor $504$
Sign $-0.678 - 0.734i$
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  + (0.619 + 1.27i)2-s + (−1.23 + 1.57i)4-s + 0.753·5-s − i·7-s + (−2.76 − 0.590i)8-s + (0.467 + 0.958i)10-s + 4.40i·11-s + 4.28i·13-s + (1.27 − 0.619i)14-s + (−0.963 − 3.88i)16-s + 7.48i·17-s + 0.157·19-s + (−0.928 + 1.18i)20-s + (−5.59 + 2.72i)22-s + 2.84·23-s + ⋯
L(s)  = 1  + (0.438 + 0.898i)2-s + (−0.616 + 0.787i)4-s + 0.337·5-s − 0.377i·7-s + (−0.977 − 0.208i)8-s + (0.147 + 0.302i)10-s + 1.32i·11-s + 1.18i·13-s + (0.339 − 0.165i)14-s + (−0.240 − 0.970i)16-s + 1.81i·17-s + 0.0360·19-s + (−0.207 + 0.265i)20-s + (−1.19 + 0.581i)22-s + 0.592·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.678 - 0.734i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.634615 + 1.44893i\)
\(L(\frac12)\) \(\approx\) \(0.634615 + 1.44893i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.619 - 1.27i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 0.753T + 5T^{2} \)
11 \( 1 - 4.40iT - 11T^{2} \)
13 \( 1 - 4.28iT - 13T^{2} \)
17 \( 1 - 7.48iT - 17T^{2} \)
19 \( 1 - 0.157T + 19T^{2} \)
23 \( 1 - 2.84T + 23T^{2} \)
29 \( 1 - 5.03T + 29T^{2} \)
31 \( 1 + 9.13iT - 31T^{2} \)
37 \( 1 + 6.38iT - 37T^{2} \)
41 \( 1 - 2.93iT - 41T^{2} \)
43 \( 1 - 1.50T + 43T^{2} \)
47 \( 1 + 6.17T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 6.17iT - 59T^{2} \)
61 \( 1 - 9.34iT - 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 2.69T + 71T^{2} \)
73 \( 1 - 1.74T + 73T^{2} \)
79 \( 1 + 12.2iT - 79T^{2} \)
83 \( 1 - 8.29iT - 83T^{2} \)
89 \( 1 + 5.13iT - 89T^{2} \)
97 \( 1 - 5.83T + 97T^{2} \)
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

−11.39554853064536069536815317126, −10.11225146503453673919304157983, −9.417658775167618347219072143775, −8.394789722194939434036356057587, −7.45256869162486548273237038960, −6.64405033723904660191605037715, −5.81187621797783957727345430048, −4.52565382456139952291038523376, −3.90586835058492296973615874746, −2.06541952012321100845908297717, 0.856979718853387166186463287217, 2.67427670521499361219391742636, 3.36512903765383213605681507792, 4.99934971086787734208303883528, 5.55407072102094432573731214449, 6.67138331901517864655547901733, 8.187420815536067749718597603562, 8.984912440634374157185343289781, 9.892976906283478211058519002149, 10.68110521749393966272872286828

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