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Properties

Label 2-48-48.11-c3-0-6
Degree $2$
Conductor $48$
Sign $0.804 - 0.594i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.56 − 1.18i)2-s + (−4.81 − 1.96i)3-s + (5.20 + 6.07i)4-s + (−6.30 + 6.30i)5-s + (10.0 + 10.7i)6-s + 24.6·7-s + (−6.17 − 21.7i)8-s + (19.3 + 18.8i)9-s + (23.6 − 8.73i)10-s + (40.4 + 40.4i)11-s + (−13.1 − 39.4i)12-s + (−47.3 + 47.3i)13-s + (−63.3 − 29.1i)14-s + (42.6 − 17.9i)15-s + (−9.86 + 63.2i)16-s − 41.7i·17-s + ⋯
L(s)  = 1  + (−0.908 − 0.418i)2-s + (−0.926 − 0.377i)3-s + (0.650 + 0.759i)4-s + (−0.563 + 0.563i)5-s + (0.683 + 0.730i)6-s + 1.33·7-s + (−0.273 − 0.961i)8-s + (0.715 + 0.698i)9-s + (0.747 − 0.276i)10-s + (1.10 + 1.10i)11-s + (−0.315 − 0.948i)12-s + (−1.00 + 1.00i)13-s + (−1.21 − 0.557i)14-s + (0.734 − 0.309i)15-s + (−0.154 + 0.988i)16-s − 0.595i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.804 - 0.594i$
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: \((2,\ 48,\ (\ :3/2),\ 0.804 - 0.594i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.621451 + 0.204694i\)
\(L(\frac12)\) \(\approx\) \(0.621451 + 0.204694i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.56 + 1.18i)T \)
3 \( 1 + (4.81 + 1.96i)T \)
good5 \( 1 + (6.30 - 6.30i)T - 125iT^{2} \)
7 \( 1 - 24.6T + 343T^{2} \)
11 \( 1 + (-40.4 - 40.4i)T + 1.33e3iT^{2} \)
13 \( 1 + (47.3 - 47.3i)T - 2.19e3iT^{2} \)
17 \( 1 + 41.7iT - 4.91e3T^{2} \)
19 \( 1 + (-10.6 - 10.6i)T + 6.85e3iT^{2} \)
23 \( 1 - 53.4iT - 1.21e4T^{2} \)
29 \( 1 + (-105. - 105. i)T + 2.43e4iT^{2} \)
31 \( 1 + 3.14iT - 2.97e4T^{2} \)
37 \( 1 + (-42.1 - 42.1i)T + 5.06e4iT^{2} \)
41 \( 1 + 152.T + 6.89e4T^{2} \)
43 \( 1 + (-221. + 221. i)T - 7.95e4iT^{2} \)
47 \( 1 - 381.T + 1.03e5T^{2} \)
53 \( 1 + (294. - 294. i)T - 1.48e5iT^{2} \)
59 \( 1 + (445. + 445. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-21.8 + 21.8i)T - 2.26e5iT^{2} \)
67 \( 1 + (572. + 572. i)T + 3.00e5iT^{2} \)
71 \( 1 - 612. iT - 3.57e5T^{2} \)
73 \( 1 + 331. iT - 3.89e5T^{2} \)
79 \( 1 - 427. iT - 4.93e5T^{2} \)
83 \( 1 + (-245. + 245. i)T - 5.71e5iT^{2} \)
89 \( 1 - 188.T + 7.04e5T^{2} \)
97 \( 1 - 1.47e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.47561721945141882904382969201, −14.26552742050003493519055859013, −12.16132348125247571944350965156, −11.75633073494532946065831501654, −10.79272637329080172591790302141, −9.387239104280609739131046959397, −7.58225740026185760789466340759, −6.92929818762965508358737187809, −4.53677181361347909741572268118, −1.71212502528242644456463510179, 0.831403935934189362623837635489, 4.68475560781677826371995358711, 6.03721782111144568750751528116, 7.74955618374135518841051891490, 8.814515679544988744281699141298, 10.35679815320995743031309200919, 11.37847562615402458695888193186, 12.17185665395609442355146023914, 14.40391979781858909860715818808, 15.33365927201137304511906418992

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