↑

Properties

Label 2-48-48.11-c1-0-5
Degree $2$
Conductor $48$
Sign $0.208 + 0.978i$
Analytic cond. $0.383281$
Root an. cond. $0.619097$
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  + (−0.416 − 1.35i)2-s + (0.966 − 1.43i)3-s + (−1.65 + 1.12i)4-s + (−1.57 + 1.57i)5-s + (−2.34 − 0.708i)6-s + 2.24·7-s + (2.20 + 1.76i)8-s + (−1.13 − 2.77i)9-s + (2.77 + 1.47i)10-s + (1.13 + 1.13i)11-s + (0.0176 + 3.46i)12-s + (−3.24 + 3.24i)13-s + (−0.935 − 3.04i)14-s + (0.739 + 3.77i)15-s + (1.47 − 3.71i)16-s + 1.66i·17-s + ⋯
L(s)  = 1  + (−0.294 − 0.955i)2-s + (0.558 − 0.829i)3-s + (−0.826 + 0.562i)4-s + (−0.702 + 0.702i)5-s + (−0.957 − 0.289i)6-s + 0.850·7-s + (0.780 + 0.624i)8-s + (−0.377 − 0.926i)9-s + (0.878 + 0.465i)10-s + (0.341 + 0.341i)11-s + (0.00510 + 0.999i)12-s + (−0.901 + 0.901i)13-s + (−0.250 − 0.812i)14-s + (0.191 + 0.975i)15-s + (0.367 − 0.929i)16-s + 0.403i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.208 + 0.978i$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1/2),\ 0.208 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.582356 - 0.471527i\)
\(L(\frac12)\) \(\approx\) \(0.582356 - 0.471527i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.416 + 1.35i)T \)
3 \( 1 + (-0.966 + 1.43i)T \)
good5 \( 1 + (1.57 - 1.57i)T - 5iT^{2} \)
7 \( 1 - 2.24T + 7T^{2} \)
11 \( 1 + (-1.13 - 1.13i)T + 11iT^{2} \)
13 \( 1 + (3.24 - 3.24i)T - 13iT^{2} \)
17 \( 1 - 1.66iT - 17T^{2} \)
19 \( 1 + (3.77 + 3.77i)T + 19iT^{2} \)
23 \( 1 + 2.26iT - 23T^{2} \)
29 \( 1 + (3.23 + 3.23i)T + 29iT^{2} \)
31 \( 1 + 1.30iT - 31T^{2} \)
37 \( 1 + (-2.30 - 2.30i)T + 37iT^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + (-3.77 + 3.77i)T - 43iT^{2} \)
47 \( 1 + 3.74T + 47T^{2} \)
53 \( 1 + (-0.972 + 0.972i)T - 53iT^{2} \)
59 \( 1 + (-3.88 - 3.88i)T + 59iT^{2} \)
61 \( 1 + (-4.19 + 4.19i)T - 61iT^{2} \)
67 \( 1 + (-8.02 - 8.02i)T + 67iT^{2} \)
71 \( 1 + 11.0iT - 71T^{2} \)
73 \( 1 - 6.38iT - 73T^{2} \)
79 \( 1 + 2.69iT - 79T^{2} \)
83 \( 1 + (-2.61 + 2.61i)T - 83iT^{2} \)
89 \( 1 + 7.35T + 89T^{2} \)
97 \( 1 + 5.67T + 97T^{2} \)
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

−14.93386674333398306069579542582, −14.31039607800830541924974555226, −12.94351548512331806788677022497, −11.83828098129248660494065931694, −11.06767496985668599272580490250, −9.370332564151671414979017907045, −8.118671092991725557645781522504, −7.06333713343041388551681232197, −4.19823136021397385132735328492, −2.29677960166614722649074286943, 4.21289449256991863971045032813, 5.36513628414595212095012969214, 7.72003282548528716475213835992, 8.425524526747607914988542948331, 9.620381904934211435696605197647, 10.98694456610594692203020077612, 12.71094722110332584301234446720, 14.27698974318256677014167699165, 14.88814190478150671465303862616, 15.95765627358636941087183709555

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