Properties

Label 2-48-16.11-c8-0-7
Degree $2$
Conductor $48$
Sign $0.397 - 0.917i$
Analytic cond. $19.5541$
Root an. cond. $4.42201$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.18 + 15.6i)2-s + (−33.0 − 33.0i)3-s + (−235. − 99.8i)4-s + (−349. − 349. i)5-s + (623. − 413. i)6-s − 4.64e3·7-s + (2.31e3 − 3.37e3i)8-s + 2.18e3i·9-s + (6.58e3 − 4.36e3i)10-s + (−549. + 549. i)11-s + (4.49e3 + 1.10e4i)12-s + (−4.12e3 + 4.12e3i)13-s + (1.47e4 − 7.27e4i)14-s + 2.31e4i·15-s + (4.55e4 + 4.70e4i)16-s + 1.21e5·17-s + ⋯
L(s)  = 1  + (−0.199 + 0.979i)2-s + (−0.408 − 0.408i)3-s + (−0.920 − 0.390i)4-s + (−0.558 − 0.558i)5-s + (0.481 − 0.318i)6-s − 1.93·7-s + (0.565 − 0.824i)8-s + 0.333i·9-s + (0.658 − 0.436i)10-s + (−0.0375 + 0.0375i)11-s + (0.216 + 0.535i)12-s + (−0.144 + 0.144i)13-s + (0.384 − 1.89i)14-s + 0.456i·15-s + (0.695 + 0.718i)16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.397 - 0.917i$
Analytic conductor: \(19.5541\)
Root analytic conductor: \(4.42201\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :4),\ 0.397 - 0.917i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.492843 + 0.323524i\)
\(L(\frac12)\) \(\approx\) \(0.492843 + 0.323524i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.18 - 15.6i)T \)
3 \( 1 + (33.0 + 33.0i)T \)
good5 \( 1 + (349. + 349. i)T + 3.90e5iT^{2} \)
7 \( 1 + 4.64e3T + 5.76e6T^{2} \)
11 \( 1 + (549. - 549. i)T - 2.14e8iT^{2} \)
13 \( 1 + (4.12e3 - 4.12e3i)T - 8.15e8iT^{2} \)
17 \( 1 - 1.21e5T + 6.97e9T^{2} \)
19 \( 1 + (-1.65e4 - 1.65e4i)T + 1.69e10iT^{2} \)
23 \( 1 + 1.04e5T + 7.83e10T^{2} \)
29 \( 1 + (7.81e5 - 7.81e5i)T - 5.00e11iT^{2} \)
31 \( 1 - 3.40e5iT - 8.52e11T^{2} \)
37 \( 1 + (-1.29e6 - 1.29e6i)T + 3.51e12iT^{2} \)
41 \( 1 + 4.04e6iT - 7.98e12T^{2} \)
43 \( 1 + (-1.64e6 + 1.64e6i)T - 1.16e13iT^{2} \)
47 \( 1 - 2.89e6iT - 2.38e13T^{2} \)
53 \( 1 + (3.17e6 + 3.17e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (5.50e6 - 5.50e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (-1.51e7 + 1.51e7i)T - 1.91e14iT^{2} \)
67 \( 1 + (1.20e7 + 1.20e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 3.40e7T + 6.45e14T^{2} \)
73 \( 1 - 3.77e7iT - 8.06e14T^{2} \)
79 \( 1 + 3.12e6iT - 1.51e15T^{2} \)
83 \( 1 + (6.95e6 + 6.95e6i)T + 2.25e15iT^{2} \)
89 \( 1 - 4.53e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.53e8T + 7.83e15T^{2} \)
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

−14.17500339405299108593727206297, −12.89301638615176493884398848720, −12.29596213268840380045659409519, −10.21239101114462005827898814957, −9.178422910844449866500254494326, −7.72345351685129506057797062810, −6.60992350514576469907327550021, −5.46585373362551263492331969437, −3.66310248923411533171904945868, −0.68407395788349277651383088972, 0.43290595409429194059675388853, 2.97088148309626964956615812715, 3.85587253051295301354709603887, 5.87444140179660299547902399597, 7.56965935213644582545799401089, 9.469196621539430583204736671770, 10.06607721295527210393841111494, 11.32007238717523419251868731227, 12.37807347465545045157759066074, 13.31452478737901574075369887086

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