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Properties

Label 2-48-16.3-c4-0-8
Degree $2$
Conductor $48$
Sign $-0.690 + 0.723i$
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  + (−3.98 − 0.368i)2-s + (−3.67 + 3.67i)3-s + (15.7 + 2.93i)4-s + (−9.21 + 9.21i)5-s + (15.9 − 13.2i)6-s + 17.0·7-s + (−61.5 − 17.4i)8-s − 27i·9-s + (40.1 − 33.3i)10-s + (−113. − 113. i)11-s + (−68.5 + 46.9i)12-s + (−184. − 184. i)13-s + (−67.9 − 6.29i)14-s − 67.7i·15-s + (238. + 92.3i)16-s − 288.·17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0921i)2-s + (−0.408 + 0.408i)3-s + (0.983 + 0.183i)4-s + (−0.368 + 0.368i)5-s + (0.444 − 0.368i)6-s + 0.348·7-s + (−0.961 − 0.273i)8-s − 0.333i·9-s + (0.401 − 0.333i)10-s + (−0.942 − 0.942i)11-s + (−0.476 + 0.326i)12-s + (−1.08 − 1.08i)13-s + (−0.346 − 0.0320i)14-s − 0.301i·15-s + (0.932 + 0.360i)16-s − 0.999·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.690 + 0.723i$
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: \((2,\ 48,\ (\ :2),\ -0.690 + 0.723i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0884259 - 0.206578i\)
\(L(\frac12)\) \(\approx\) \(0.0884259 - 0.206578i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.98 + 0.368i)T \)
3 \( 1 + (3.67 - 3.67i)T \)
good5 \( 1 + (9.21 - 9.21i)T - 625iT^{2} \)
7 \( 1 - 17.0T + 2.40e3T^{2} \)
11 \( 1 + (113. + 113. i)T + 1.46e4iT^{2} \)
13 \( 1 + (184. + 184. i)T + 2.85e4iT^{2} \)
17 \( 1 + 288.T + 8.35e4T^{2} \)
19 \( 1 + (-271. + 271. i)T - 1.30e5iT^{2} \)
23 \( 1 - 230.T + 2.79e5T^{2} \)
29 \( 1 + (566. + 566. i)T + 7.07e5iT^{2} \)
31 \( 1 + 429. iT - 9.23e5T^{2} \)
37 \( 1 + (1.08e3 - 1.08e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.36e3iT - 2.82e6T^{2} \)
43 \( 1 + (949. + 949. i)T + 3.41e6iT^{2} \)
47 \( 1 + 1.04e3iT - 4.87e6T^{2} \)
53 \( 1 + (-476. + 476. i)T - 7.89e6iT^{2} \)
59 \( 1 + (-3.14e3 - 3.14e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (2.61e3 + 2.61e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (3.50e3 - 3.50e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 8.35e3T + 2.54e7T^{2} \)
73 \( 1 - 8.22e3iT - 2.83e7T^{2} \)
79 \( 1 + 6.26e3iT - 3.89e7T^{2} \)
83 \( 1 + (8.29e3 - 8.29e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 4.21e3iT - 6.27e7T^{2} \)
97 \( 1 - 2.53e3T + 8.85e7T^{2} \)
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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

−15.09681215213566322795784039270, −13.14520614512352778747209550126, −11.57925715412290843136700005018, −10.89518457627442535982349433923, −9.779067951310119378112380602458, −8.309015438018036823475948094881, −7.14919419198494303096650193127, −5.37717573748334336954963072267, −2.94493776720341750657334315892, −0.18720686391032189991483306203, 1.99062293373930058158132230776, 5.00342896377649741216921017355, 6.90025298339831152790125033339, 7.82599853928167637285431983299, 9.246200445395733582469289605447, 10.54057649133469587900630139562, 11.76768324425591899213075740879, 12.60398742848607749917512264880, 14.41701312271380995096688343568, 15.66926478717714606842854125129

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