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Properties

Label 2-48-3.2-c4-0-3
Degree $2$
Conductor $48$
Sign $0.912 - 0.409i$
Analytic cond. $4.96175$
Root an. cond. $2.22750$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.21 − 3.68i)3-s + 44.7i·5-s + 37.2·7-s + (53.8 − 60.5i)9-s − 24.1i·11-s + 111.·13-s + (164. + 367. i)15-s + 92.5i·17-s − 249.·19-s + (306. − 137. i)21-s − 671. i·23-s − 1.37e3·25-s + (219. − 695. i)27-s + 311. i·29-s − 1.42e3·31-s + ⋯
L(s)  = 1  + (0.912 − 0.409i)3-s + 1.78i·5-s + 0.760·7-s + (0.664 − 0.747i)9-s − 0.199i·11-s + 0.657·13-s + (0.732 + 1.63i)15-s + 0.320i·17-s − 0.690·19-s + (0.693 − 0.311i)21-s − 1.26i·23-s − 2.20·25-s + (0.300 − 0.953i)27-s + 0.369i·29-s − 1.48·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.912 - 0.409i$
Analytic conductor: \(4.96175\)
Root analytic conductor: \(2.22750\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :2),\ 0.912 - 0.409i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.05506 + 0.439978i\)
\(L(\frac12)\) \(\approx\) \(2.05506 + 0.439978i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-8.21 + 3.68i)T \)
good5 \( 1 - 44.7iT - 625T^{2} \)
7 \( 1 - 37.2T + 2.40e3T^{2} \)
11 \( 1 + 24.1iT - 1.46e4T^{2} \)
13 \( 1 - 111.T + 2.85e4T^{2} \)
17 \( 1 - 92.5iT - 8.35e4T^{2} \)
19 \( 1 + 249.T + 1.30e5T^{2} \)
23 \( 1 + 671. iT - 2.79e5T^{2} \)
29 \( 1 - 311. iT - 7.07e5T^{2} \)
31 \( 1 + 1.42e3T + 9.23e5T^{2} \)
37 \( 1 - 509.T + 1.87e6T^{2} \)
41 \( 1 + 2.59e3iT - 2.82e6T^{2} \)
43 \( 1 - 544.T + 3.41e6T^{2} \)
47 \( 1 - 985. iT - 4.87e6T^{2} \)
53 \( 1 - 897. iT - 7.89e6T^{2} \)
59 \( 1 + 3.79e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.04e3T + 1.38e7T^{2} \)
67 \( 1 - 3.56e3T + 2.01e7T^{2} \)
71 \( 1 - 5.33e3iT - 2.54e7T^{2} \)
73 \( 1 + 698.T + 2.83e7T^{2} \)
79 \( 1 + 5.74e3T + 3.89e7T^{2} \)
83 \( 1 + 8.00e3iT - 4.74e7T^{2} \)
89 \( 1 + 6.50e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.24e4T + 8.85e7T^{2} \)
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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

−14.49023592350654926372179339104, −14.37006042940753231332005766219, −12.88287785239410292992735664689, −11.24965055579947798048720553663, −10.37366436170507231714683233460, −8.675674166815932750244584319780, −7.46260031816399037906513065234, −6.36263046929839813078395454966, −3.70164229403187575958540404362, −2.22450807919906791099545897465, 1.58444309195845848303725559886, 4.11000853571574819028106162533, 5.25879470161811167342838583969, 7.83262603731394397823167542772, 8.710630804886119223757142491383, 9.629811636406627527339393698583, 11.34633685640111965079704425127, 12.80288225414885775639699623928, 13.57639990115325107612054565114, 14.88577633454441217216582686265

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