↑

Properties

Label 2-48-48.5-c2-0-4
Degree $2$
Conductor $48$
Sign $0.170 - 0.985i$
Analytic cond. $1.30790$
Root an. cond. $1.14363$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.312 + 1.97i)2-s + (2.75 + 1.18i)3-s + (−3.80 + 1.23i)4-s + (0.00985 − 0.00985i)5-s + (−1.48 + 5.81i)6-s − 6.42i·7-s + (−3.62 − 7.13i)8-s + (6.19 + 6.53i)9-s + (0.0225 + 0.0163i)10-s + (−9.07 + 9.07i)11-s + (−11.9 − 1.10i)12-s + (12.6 − 12.6i)13-s + (12.6 − 2.00i)14-s + (0.0388 − 0.0154i)15-s + (12.9 − 9.39i)16-s − 19.0i·17-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (0.918 + 0.395i)3-s + (−0.951 + 0.308i)4-s + (0.00197 − 0.00197i)5-s + (−0.246 + 0.969i)6-s − 0.917i·7-s + (−0.453 − 0.891i)8-s + (0.687 + 0.725i)9-s + (0.00225 + 0.00163i)10-s + (−0.824 + 0.824i)11-s + (−0.995 − 0.0923i)12-s + (0.969 − 0.969i)13-s + (0.906 − 0.143i)14-s + (0.00259 − 0.00103i)15-s + (0.809 − 0.586i)16-s − 1.11i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.170 - 0.985i$
Analytic conductor: \(1.30790\)
Root analytic conductor: \(1.14363\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ 0.170 - 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.06052 + 0.892557i\)
\(L(\frac12)\) \(\approx\) \(1.06052 + 0.892557i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.312 - 1.97i)T \)
3 \( 1 + (-2.75 - 1.18i)T \)
good5 \( 1 + (-0.00985 + 0.00985i)T - 25iT^{2} \)
7 \( 1 + 6.42iT - 49T^{2} \)
11 \( 1 + (9.07 - 9.07i)T - 121iT^{2} \)
13 \( 1 + (-12.6 + 12.6i)T - 169iT^{2} \)
17 \( 1 + 19.0iT - 289T^{2} \)
19 \( 1 + (2.07 - 2.07i)T - 361iT^{2} \)
23 \( 1 + 19.5T + 529T^{2} \)
29 \( 1 + (-11.1 - 11.1i)T + 841iT^{2} \)
31 \( 1 + 59.9T + 961T^{2} \)
37 \( 1 + (-9.32 - 9.32i)T + 1.36e3iT^{2} \)
41 \( 1 + 47.2T + 1.68e3T^{2} \)
43 \( 1 + (-24.1 - 24.1i)T + 1.84e3iT^{2} \)
47 \( 1 + 6.29iT - 2.20e3T^{2} \)
53 \( 1 + (20.6 - 20.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (-60.3 + 60.3i)T - 3.48e3iT^{2} \)
61 \( 1 + (-48.0 + 48.0i)T - 3.72e3iT^{2} \)
67 \( 1 + (23.7 - 23.7i)T - 4.48e3iT^{2} \)
71 \( 1 + 13.5T + 5.04e3T^{2} \)
73 \( 1 + 31.4iT - 5.32e3T^{2} \)
79 \( 1 - 47.4T + 6.24e3T^{2} \)
83 \( 1 + (-70.3 - 70.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 95.1T + 7.92e3T^{2} \)
97 \( 1 - 61.6T + 9.40e3T^{2} \)
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

−15.65195419808528734949337536806, −14.65906920933887274654637409653, −13.60018285662308775270155726789, −12.89674263829413214146481208244, −10.56635651524454623320915603671, −9.465484590076420080518627334083, −8.063302122908714699208123152679, −7.20649707332244917077759988149, −5.12568898404102961979249161886, −3.60935023540904945344208377328, 2.14946129582601885156194617867, 3.79371961092650800389969281347, 5.96319283527627198099678657733, 8.295313908240820392004831680352, 8.983797314224881975755273085307, 10.47516788106905051595142421448, 11.83299274816307046033094557443, 12.91756081880385980298571481619, 13.79882890634821464817461065480, 14.82417970027163931723844974393

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