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Properties

Label 2-48-16.13-c3-0-7
Degree $2$
Conductor $48$
Sign $0.999 + 0.0172i$
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  + (2.70 + 0.832i)2-s + (−2.12 − 2.12i)3-s + (6.61 + 4.50i)4-s + (11.7 − 11.7i)5-s + (−3.96 − 7.50i)6-s + 12.5i·7-s + (14.1 + 17.6i)8-s + 8.99i·9-s + (41.6 − 22.0i)10-s + (−17.0 + 17.0i)11-s + (−4.47 − 23.5i)12-s + (−49.2 − 49.2i)13-s + (−10.4 + 33.9i)14-s − 50.0·15-s + (23.4 + 59.5i)16-s − 51.8·17-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)2-s + (−0.408 − 0.408i)3-s + (0.826 + 0.562i)4-s + (1.05 − 1.05i)5-s + (−0.269 − 0.510i)6-s + 0.679i·7-s + (0.624 + 0.781i)8-s + 0.333i·9-s + (1.31 − 0.697i)10-s + (−0.466 + 0.466i)11-s + (−0.107 − 0.567i)12-s + (−1.05 − 1.05i)13-s + (−0.199 + 0.648i)14-s − 0.861·15-s + (0.366 + 0.930i)16-s − 0.739·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0172i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.999 + 0.0172i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ 0.999 + 0.0172i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.19841 - 0.0189501i\)
\(L(\frac12)\) \(\approx\) \(2.19841 - 0.0189501i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.70 - 0.832i)T \)
3 \( 1 + (2.12 + 2.12i)T \)
good5 \( 1 + (-11.7 + 11.7i)T - 125iT^{2} \)
7 \( 1 - 12.5iT - 343T^{2} \)
11 \( 1 + (17.0 - 17.0i)T - 1.33e3iT^{2} \)
13 \( 1 + (49.2 + 49.2i)T + 2.19e3iT^{2} \)
17 \( 1 + 51.8T + 4.91e3T^{2} \)
19 \( 1 + (11.6 + 11.6i)T + 6.85e3iT^{2} \)
23 \( 1 - 74.5iT - 1.21e4T^{2} \)
29 \( 1 + (-211. - 211. i)T + 2.43e4iT^{2} \)
31 \( 1 + 326.T + 2.97e4T^{2} \)
37 \( 1 + (-110. + 110. i)T - 5.06e4iT^{2} \)
41 \( 1 + 348. iT - 6.89e4T^{2} \)
43 \( 1 + (-205. + 205. i)T - 7.95e4iT^{2} \)
47 \( 1 - 254.T + 1.03e5T^{2} \)
53 \( 1 + (225. - 225. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-285. + 285. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-286. - 286. i)T + 2.26e5iT^{2} \)
67 \( 1 + (627. + 627. i)T + 3.00e5iT^{2} \)
71 \( 1 + 274. iT - 3.57e5T^{2} \)
73 \( 1 + 298. iT - 3.89e5T^{2} \)
79 \( 1 + 175.T + 4.93e5T^{2} \)
83 \( 1 + (-125. - 125. i)T + 5.71e5iT^{2} \)
89 \( 1 - 900. iT - 7.04e5T^{2} \)
97 \( 1 - 5.27T + 9.12e5T^{2} \)
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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

−15.12743930603740872930803863562, −13.79270304476331236588938189802, −12.66989269973618224791061456444, −12.40476530488787865276845690961, −10.62348198208644885577170445231, −8.959043456908851106351359500755, −7.34144625178942130654434304871, −5.68921650712228447723601379311, −5.03137298737696355824324681843, −2.23130572978299042511475581909, 2.51486336640243487312565473750, 4.43853898119637296293849655442, 6.03515835304549882214921734738, 7.02461419545144466168223969857, 9.758990590998877241403553410467, 10.57905470852113818186109600532, 11.51652142295962159391936624561, 13.08044051419644036879867996524, 14.10433693684811219754800957929, 14.76038270887027542932032213821

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