↑

Properties

Label 2-2808-2808.1507-c0-0-5
Degree $2$
Conductor $2808$
Sign $0.230 + 0.973i$
Analytic cond. $1.40137$
Root an. cond. $1.18379$
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  + (0.766 − 0.642i)2-s + (−0.0581 + 0.998i)3-s + (0.173 − 0.984i)4-s + (1.57 − 0.571i)5-s + (0.597 + 0.802i)6-s + (−0.344 − 1.95i)7-s + (−0.500 − 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.835 − 1.44i)10-s + (0.973 + 0.230i)12-s + (0.766 + 0.642i)13-s + (−1.52 − 1.27i)14-s + (0.479 + 1.60i)15-s + (−0.939 − 0.342i)16-s + (−0.973 + 1.68i)17-s + (−0.835 + 0.549i)18-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.0581 + 0.998i)3-s + (0.173 − 0.984i)4-s + (1.57 − 0.571i)5-s + (0.597 + 0.802i)6-s + (−0.344 − 1.95i)7-s + (−0.500 − 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.835 − 1.44i)10-s + (0.973 + 0.230i)12-s + (0.766 + 0.642i)13-s + (−1.52 − 1.27i)14-s + (0.479 + 1.60i)15-s + (−0.939 − 0.342i)16-s + (−0.973 + 1.68i)17-s + (−0.835 + 0.549i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(1.40137\)
Root analytic conductor: \(1.18379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (1507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2808,\ (\ :0),\ 0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.198423980\)
\(L(\frac12)\) \(\approx\) \(2.198423980\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (0.0581 - 0.998i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (-1.57 + 0.571i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.344 + 1.95i)T + (-0.939 + 0.342i)T^{2} \)
11 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.973 - 1.68i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
37 \( 1 + (-0.0581 + 0.100i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-1.28 - 0.469i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.0996 + 0.564i)T + (-0.939 + 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.597 - 1.03i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.766 - 0.642i)T^{2} \)
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

−9.214762058554670344348274194939, −8.384147412986509786704604715044, −6.89093460438609377198254694268, −6.20513269164956691068052395895, −5.72002995305639610216949062659, −4.54299728566390525639188072012, −4.21658778266271182056426465961, −3.44247826820521996524265042661, −2.13043890737128858105094776602, −1.15304403333182564749535209697, 1.96631244457249156645052925446, 2.65890538929104198978692333168, 3.05389388718112398790661842465, 4.98420616083098368686106768072, 5.58888615365451877683901221214, 6.08120112238996891299459853374, 6.55615967618372082443621880976, 7.33012391680095690223819222082, 8.388503388363622567754421476526, 9.026612446436332025796169212335

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