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Properties

Label 2-48-16.3-c4-0-10
Degree $2$
Conductor $48$
Sign $0.850 + 0.526i$
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  + (3.75 − 1.38i)2-s + (−3.67 + 3.67i)3-s + (12.1 − 10.3i)4-s + (13.5 − 13.5i)5-s + (−8.72 + 18.8i)6-s + 44.0·7-s + (31.4 − 55.7i)8-s − 27i·9-s + (32.1 − 69.4i)10-s + (61.0 + 61.0i)11-s + (−6.70 + 82.8i)12-s + (3.87 + 3.87i)13-s + (165. − 60.7i)14-s + 99.4i·15-s + (41.1 − 252. i)16-s − 519.·17-s + ⋯
L(s)  = 1  + (0.938 − 0.345i)2-s + (−0.408 + 0.408i)3-s + (0.761 − 0.647i)4-s + (0.541 − 0.541i)5-s + (−0.242 + 0.524i)6-s + 0.898·7-s + (0.491 − 0.870i)8-s − 0.333i·9-s + (0.321 − 0.694i)10-s + (0.504 + 0.504i)11-s + (−0.0465 + 0.575i)12-s + (0.0229 + 0.0229i)13-s + (0.843 − 0.310i)14-s + 0.441i·15-s + (0.160 − 0.986i)16-s − 1.79·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.850 + 0.526i$
Analytic conductor: \(4.96175\)
Root analytic conductor: \(2.22750\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :2),\ 0.850 + 0.526i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.47960 - 0.705183i\)
\(L(\frac12)\) \(\approx\) \(2.47960 - 0.705183i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.75 + 1.38i)T \)
3 \( 1 + (3.67 - 3.67i)T \)
good5 \( 1 + (-13.5 + 13.5i)T - 625iT^{2} \)
7 \( 1 - 44.0T + 2.40e3T^{2} \)
11 \( 1 + (-61.0 - 61.0i)T + 1.46e4iT^{2} \)
13 \( 1 + (-3.87 - 3.87i)T + 2.85e4iT^{2} \)
17 \( 1 + 519.T + 8.35e4T^{2} \)
19 \( 1 + (229. - 229. i)T - 1.30e5iT^{2} \)
23 \( 1 - 100.T + 2.79e5T^{2} \)
29 \( 1 + (-88.1 - 88.1i)T + 7.07e5iT^{2} \)
31 \( 1 - 578. iT - 9.23e5T^{2} \)
37 \( 1 + (1.63e3 - 1.63e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 3.27e3iT - 2.82e6T^{2} \)
43 \( 1 + (-1.87e3 - 1.87e3i)T + 3.41e6iT^{2} \)
47 \( 1 + 1.76e3iT - 4.87e6T^{2} \)
53 \( 1 + (-1.22e3 + 1.22e3i)T - 7.89e6iT^{2} \)
59 \( 1 + (-2.72e3 - 2.72e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-585. - 585. i)T + 1.38e7iT^{2} \)
67 \( 1 + (-4.51e3 + 4.51e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 9.42e3T + 2.54e7T^{2} \)
73 \( 1 + 3.93e3iT - 2.83e7T^{2} \)
79 \( 1 - 7.53e3iT - 3.89e7T^{2} \)
83 \( 1 + (1.77e3 - 1.77e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 950. iT - 6.27e7T^{2} \)
97 \( 1 - 1.55e3T + 8.85e7T^{2} \)
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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

−14.73524588954802118576015604011, −13.61866462115446392586326034502, −12.48260579093853569287702945010, −11.39703926675898096220592400181, −10.37593137133366081956027554123, −8.909926578382206166327448193373, −6.75131888651279488829259613968, −5.28784662334197079574423711190, −4.25335666724142862861978555030, −1.77841695931862011545934715468, 2.24653449846956889040009070210, 4.49519974598024427180822641989, 6.02753963739611689031627260986, 7.01732383240933831508413278024, 8.581711126598247791089781016340, 10.84813249098607016661850978578, 11.49573182538744747912225858626, 12.94168109106752179535991758076, 13.88184932090019998414662555628, 14.76694868613907232155368274802

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