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Properties

Label 2-48-48.11-c3-0-19
Degree $2$
Conductor $48$
Sign $0.113 + 0.993i$
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  + (0.763 − 2.72i)2-s + (5.19 − 0.151i)3-s + (−6.83 − 4.16i)4-s + (4.66 − 4.66i)5-s + (3.55 − 14.2i)6-s + 0.405·7-s + (−16.5 + 15.4i)8-s + (26.9 − 1.56i)9-s + (−9.14 − 16.2i)10-s + (−5.82 − 5.82i)11-s + (−36.1 − 20.5i)12-s + (−35.2 + 35.2i)13-s + (0.309 − 1.10i)14-s + (23.5 − 24.9i)15-s + (29.3 + 56.8i)16-s − 49.3i·17-s + ⋯
L(s)  = 1  + (0.270 − 0.962i)2-s + (0.999 − 0.0290i)3-s + (−0.854 − 0.520i)4-s + (0.417 − 0.417i)5-s + (0.241 − 0.970i)6-s + 0.0219·7-s + (−0.731 + 0.681i)8-s + (0.998 − 0.0581i)9-s + (−0.289 − 0.514i)10-s + (−0.159 − 0.159i)11-s + (−0.868 − 0.495i)12-s + (−0.751 + 0.751i)13-s + (0.00591 − 0.0210i)14-s + (0.405 − 0.429i)15-s + (0.459 + 0.888i)16-s − 0.703i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.113 + 0.993i$
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: \((2,\ 48,\ (\ :3/2),\ 0.113 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.43504 - 1.28100i\)
\(L(\frac12)\) \(\approx\) \(1.43504 - 1.28100i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.763 + 2.72i)T \)
3 \( 1 + (-5.19 + 0.151i)T \)
good5 \( 1 + (-4.66 + 4.66i)T - 125iT^{2} \)
7 \( 1 - 0.405T + 343T^{2} \)
11 \( 1 + (5.82 + 5.82i)T + 1.33e3iT^{2} \)
13 \( 1 + (35.2 - 35.2i)T - 2.19e3iT^{2} \)
17 \( 1 + 49.3iT - 4.91e3T^{2} \)
19 \( 1 + (-108. - 108. i)T + 6.85e3iT^{2} \)
23 \( 1 - 130. iT - 1.21e4T^{2} \)
29 \( 1 + (172. + 172. i)T + 2.43e4iT^{2} \)
31 \( 1 + 36.1iT - 2.97e4T^{2} \)
37 \( 1 + (257. + 257. i)T + 5.06e4iT^{2} \)
41 \( 1 + 5.87T + 6.89e4T^{2} \)
43 \( 1 + (-170. + 170. i)T - 7.95e4iT^{2} \)
47 \( 1 + 181.T + 1.03e5T^{2} \)
53 \( 1 + (148. - 148. i)T - 1.48e5iT^{2} \)
59 \( 1 + (567. + 567. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-481. + 481. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-296. - 296. i)T + 3.00e5iT^{2} \)
71 \( 1 - 533. iT - 3.57e5T^{2} \)
73 \( 1 - 178. iT - 3.89e5T^{2} \)
79 \( 1 - 528. iT - 4.93e5T^{2} \)
83 \( 1 + (-713. + 713. i)T - 5.71e5iT^{2} \)
89 \( 1 - 204.T + 7.04e5T^{2} \)
97 \( 1 - 275.T + 9.12e5T^{2} \)
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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

−14.38510423192457414873443321408, −13.76587364992118821626843736824, −12.70296566421590242772368270959, −11.53242225668048921272015530535, −9.774635163476749470622413052037, −9.296706571919606244939240079119, −7.66976681548948934636675557704, −5.29899900323881701655813865662, −3.59878182753573292425028081027, −1.83924860026765039050600761976, 3.03857469378377731092670161126, 4.93131154801195101403233031225, 6.75995357207590190693217319255, 7.87379324788344464940872202504, 9.130197991610662616759271146920, 10.28012444601289635050442313633, 12.50979865239024153449626360788, 13.47507299234697229039207680141, 14.48023476841495496730781549707, 15.16902894525375141160640717429

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