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Properties

Label 2-48-16.5-c3-0-11
Degree $2$
Conductor $48$
Sign $-0.969 + 0.245i$
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  + (−0.987 − 2.65i)2-s + (2.12 − 2.12i)3-s + (−6.05 + 5.23i)4-s + (−11.8 − 11.8i)5-s + (−7.71 − 3.52i)6-s − 0.485i·7-s + (19.8 + 10.8i)8-s − 8.99i·9-s + (−19.7 + 43.2i)10-s + (−30.9 − 30.9i)11-s + (−1.73 + 23.9i)12-s + (−18.4 + 18.4i)13-s + (−1.28 + 0.478i)14-s − 50.4·15-s + (9.24 − 63.3i)16-s + 135.·17-s + ⋯
L(s)  = 1  + (−0.348 − 0.937i)2-s + (0.408 − 0.408i)3-s + (−0.756 + 0.654i)4-s + (−1.06 − 1.06i)5-s + (−0.525 − 0.240i)6-s − 0.0261i·7-s + (0.876 + 0.480i)8-s − 0.333i·9-s + (−0.625 + 1.36i)10-s + (−0.848 − 0.848i)11-s + (−0.0418 + 0.575i)12-s + (−0.393 + 0.393i)13-s + (−0.0245 + 0.00913i)14-s − 0.868·15-s + (0.144 − 0.989i)16-s + 1.93·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.969 + 0.245i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.969 + 0.245i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.105598 - 0.848036i\)
\(L(\frac12)\) \(\approx\) \(0.105598 - 0.848036i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.987 + 2.65i)T \)
3 \( 1 + (-2.12 + 2.12i)T \)
good5 \( 1 + (11.8 + 11.8i)T + 125iT^{2} \)
7 \( 1 + 0.485iT - 343T^{2} \)
11 \( 1 + (30.9 + 30.9i)T + 1.33e3iT^{2} \)
13 \( 1 + (18.4 - 18.4i)T - 2.19e3iT^{2} \)
17 \( 1 - 135.T + 4.91e3T^{2} \)
19 \( 1 + (-65.8 + 65.8i)T - 6.85e3iT^{2} \)
23 \( 1 + 128. iT - 1.21e4T^{2} \)
29 \( 1 + (-6.64 + 6.64i)T - 2.43e4iT^{2} \)
31 \( 1 + 15.1T + 2.97e4T^{2} \)
37 \( 1 + (-51.4 - 51.4i)T + 5.06e4iT^{2} \)
41 \( 1 - 410. iT - 6.89e4T^{2} \)
43 \( 1 + (-69.9 - 69.9i)T + 7.95e4iT^{2} \)
47 \( 1 + 487.T + 1.03e5T^{2} \)
53 \( 1 + (217. + 217. i)T + 1.48e5iT^{2} \)
59 \( 1 + (293. + 293. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-207. + 207. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-284. + 284. i)T - 3.00e5iT^{2} \)
71 \( 1 - 614. iT - 3.57e5T^{2} \)
73 \( 1 - 486. iT - 3.89e5T^{2} \)
79 \( 1 - 960.T + 4.93e5T^{2} \)
83 \( 1 + (-463. + 463. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.27e3iT - 7.04e5T^{2} \)
97 \( 1 + 994.T + 9.12e5T^{2} \)
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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

−14.35775294754145409454105382426, −13.09745019207039843711895061428, −12.27663273529408127941419664379, −11.34893853850718707037356484901, −9.729963745312137024038126635148, −8.408529399511264928667591913206, −7.74453018355791605387131426259, −4.89477892226352064745948975228, −3.20412086605477390798507592002, −0.74975134496162749860328636908, 3.53449507208014420266043670009, 5.38501224850758983232648874951, 7.46431480071737310064691004228, 7.79538820190796973048656061741, 9.717402828204906082732004496706, 10.55314701983827039762430300204, 12.23313171087944658663803112070, 13.97546634127273360763490435963, 14.90196656215938214689017137763, 15.49167361899917733425893422717

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