↑

Properties

Label 2-48-48.11-c3-0-7
Degree $2$
Conductor $48$
Sign $-0.144 - 0.989i$
Analytic cond. $2.83209$
Root an. cond. $1.68288$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.53 + 1.25i)2-s + (−3.58 + 3.76i)3-s + (4.83 + 6.37i)4-s + (−4.71 + 4.71i)5-s + (−13.8 + 5.03i)6-s + 4.67·7-s + (4.23 + 22.2i)8-s + (−1.36 − 26.9i)9-s + (−17.8 + 6.01i)10-s + (29.7 + 29.7i)11-s + (−41.3 − 4.60i)12-s + (36.9 − 36.9i)13-s + (11.8 + 5.88i)14-s + (−0.874 − 34.6i)15-s + (−17.2 + 61.6i)16-s − 109. i·17-s + ⋯
L(s)  = 1  + (0.895 + 0.444i)2-s + (−0.689 + 0.724i)3-s + (0.604 + 0.796i)4-s + (−0.421 + 0.421i)5-s + (−0.939 + 0.342i)6-s + 0.252·7-s + (0.187 + 0.982i)8-s + (−0.0504 − 0.998i)9-s + (−0.565 + 0.190i)10-s + (0.815 + 0.815i)11-s + (−0.993 − 0.110i)12-s + (0.788 − 0.788i)13-s + (0.226 + 0.112i)14-s + (−0.0150 − 0.596i)15-s + (−0.269 + 0.963i)16-s − 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.144 - 0.989i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.144 - 0.989i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.14670 + 1.32681i\)
\(L(\frac12)\) \(\approx\) \(1.14670 + 1.32681i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.53 - 1.25i)T \)
3 \( 1 + (3.58 - 3.76i)T \)
good5 \( 1 + (4.71 - 4.71i)T - 125iT^{2} \)
7 \( 1 - 4.67T + 343T^{2} \)
11 \( 1 + (-29.7 - 29.7i)T + 1.33e3iT^{2} \)
13 \( 1 + (-36.9 + 36.9i)T - 2.19e3iT^{2} \)
17 \( 1 + 109. iT - 4.91e3T^{2} \)
19 \( 1 + (-28.6 - 28.6i)T + 6.85e3iT^{2} \)
23 \( 1 - 0.193iT - 1.21e4T^{2} \)
29 \( 1 + (162. + 162. i)T + 2.43e4iT^{2} \)
31 \( 1 + 179. iT - 2.97e4T^{2} \)
37 \( 1 + (-194. - 194. i)T + 5.06e4iT^{2} \)
41 \( 1 + 49.2T + 6.89e4T^{2} \)
43 \( 1 + (336. - 336. i)T - 7.95e4iT^{2} \)
47 \( 1 - 187.T + 1.03e5T^{2} \)
53 \( 1 + (-195. + 195. i)T - 1.48e5iT^{2} \)
59 \( 1 + (302. + 302. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-501. + 501. i)T - 2.26e5iT^{2} \)
67 \( 1 + (36.4 + 36.4i)T + 3.00e5iT^{2} \)
71 \( 1 + 637. iT - 3.57e5T^{2} \)
73 \( 1 + 90.3iT - 3.89e5T^{2} \)
79 \( 1 + 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 + (256. - 256. i)T - 5.71e5iT^{2} \)
89 \( 1 - 818.T + 7.04e5T^{2} \)
97 \( 1 - 667.T + 9.12e5T^{2} \)
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

−15.32728853073515192990029074781, −14.70403667084601489521871381630, −13.24820839774369050542509340437, −11.77083910151822035857915275998, −11.27209571059664535117661250404, −9.603080356008737287409429948286, −7.66929692433992982955359246202, −6.31138141697934057493206153150, −4.90307129655302566451362711663, −3.54132441586860453570353108674, 1.39770481421017432742687013316, 4.00781277999279662281945612516, 5.66254480267918079790527682591, 6.81354620624778883361357064833, 8.615996150216632230649329603556, 10.73468875245367161956484554352, 11.55421869056625626965030078394, 12.47244048550022460086711828039, 13.50545594129917369165002635820, 14.53179784922898907624112496117

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