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Properties

Label 2-48-48.29-c2-0-1
Degree $2$
Conductor $48$
Sign $-0.801 - 0.597i$
Analytic cond. $1.30790$
Root an. cond. $1.14363$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.312 + 1.97i)2-s + (−1.18 + 2.75i)3-s + (−3.80 − 1.23i)4-s + (−0.00985 − 0.00985i)5-s + (−5.07 − 3.20i)6-s + 6.42i·7-s + (3.62 − 7.13i)8-s + (−6.19 − 6.53i)9-s + (0.0225 − 0.0163i)10-s + (9.07 + 9.07i)11-s + (7.90 − 9.02i)12-s + (12.6 + 12.6i)13-s + (−12.6 − 2.00i)14-s + (0.0388 − 0.0154i)15-s + (12.9 + 9.39i)16-s − 19.0i·17-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (−0.395 + 0.918i)3-s + (−0.951 − 0.308i)4-s + (−0.00197 − 0.00197i)5-s + (−0.845 − 0.533i)6-s + 0.917i·7-s + (0.453 − 0.891i)8-s + (−0.687 − 0.725i)9-s + (0.00225 − 0.00163i)10-s + (0.824 + 0.824i)11-s + (0.659 − 0.752i)12-s + (0.969 + 0.969i)13-s + (−0.906 − 0.143i)14-s + (0.00259 − 0.00103i)15-s + (0.809 + 0.586i)16-s − 1.11i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.597i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.801 - 0.597i$
Analytic conductor: \(1.30790\)
Root analytic conductor: \(1.14363\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ -0.801 - 0.597i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.266585 + 0.803661i\)
\(L(\frac12)\) \(\approx\) \(0.266585 + 0.803661i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.312 - 1.97i)T \)
3 \( 1 + (1.18 - 2.75i)T \)
good5 \( 1 + (0.00985 + 0.00985i)T + 25iT^{2} \)
7 \( 1 - 6.42iT - 49T^{2} \)
11 \( 1 + (-9.07 - 9.07i)T + 121iT^{2} \)
13 \( 1 + (-12.6 - 12.6i)T + 169iT^{2} \)
17 \( 1 + 19.0iT - 289T^{2} \)
19 \( 1 + (2.07 + 2.07i)T + 361iT^{2} \)
23 \( 1 - 19.5T + 529T^{2} \)
29 \( 1 + (11.1 - 11.1i)T - 841iT^{2} \)
31 \( 1 + 59.9T + 961T^{2} \)
37 \( 1 + (-9.32 + 9.32i)T - 1.36e3iT^{2} \)
41 \( 1 - 47.2T + 1.68e3T^{2} \)
43 \( 1 + (-24.1 + 24.1i)T - 1.84e3iT^{2} \)
47 \( 1 + 6.29iT - 2.20e3T^{2} \)
53 \( 1 + (-20.6 - 20.6i)T + 2.80e3iT^{2} \)
59 \( 1 + (60.3 + 60.3i)T + 3.48e3iT^{2} \)
61 \( 1 + (-48.0 - 48.0i)T + 3.72e3iT^{2} \)
67 \( 1 + (23.7 + 23.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 13.5T + 5.04e3T^{2} \)
73 \( 1 - 31.4iT - 5.32e3T^{2} \)
79 \( 1 - 47.4T + 6.24e3T^{2} \)
83 \( 1 + (70.3 - 70.3i)T - 6.88e3iT^{2} \)
89 \( 1 + 95.1T + 7.92e3T^{2} \)
97 \( 1 - 61.6T + 9.40e3T^{2} \)
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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

−15.96256157647489616915275997459, −14.97282108448869728440245408157, −14.12741715415736655852496511718, −12.40327467795031509742329998782, −11.09329032108650004020231477306, −9.396540882046507336891443155219, −8.942621051775435595681968436573, −6.88468592448154371756917519980, −5.58554247081153905574436201127, −4.20715559226518889787013539308, 1.15046671920281075487615447417, 3.63467062135958523872414921995, 5.83441062958438635150373741278, 7.62796547585578633328886721564, 8.876314404757390408678642535872, 10.73580811627619523803675900844, 11.24055065157192006551637310568, 12.79024717792852324260733411302, 13.34644916474894238331882980637, 14.52104221874584671224828701265

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