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Properties

Label 2-48-48.29-c2-0-13
Degree $2$
Conductor $48$
Sign $-0.998 + 0.0569i$
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  + (−0.420 − 1.95i)2-s + (−2.77 + 1.13i)3-s + (−3.64 + 1.64i)4-s + (−6.28 − 6.28i)5-s + (3.38 + 4.95i)6-s − 1.64i·7-s + (4.75 + 6.43i)8-s + (6.43 − 6.29i)9-s + (−9.64 + 14.9i)10-s + (4.75 + 4.75i)11-s + (8.26 − 8.70i)12-s + (−9.35 − 9.35i)13-s + (−3.21 + 0.692i)14-s + (24.5 + 10.3i)15-s + (10.5 − 12i)16-s − 11.4i·17-s + ⋯
L(s)  = 1  + (−0.210 − 0.977i)2-s + (−0.926 + 0.377i)3-s + (−0.911 + 0.411i)4-s + (−1.25 − 1.25i)5-s + (0.563 + 0.825i)6-s − 0.235i·7-s + (0.594 + 0.804i)8-s + (0.715 − 0.699i)9-s + (−0.964 + 1.49i)10-s + (0.432 + 0.432i)11-s + (0.688 − 0.725i)12-s + (−0.719 − 0.719i)13-s + (−0.229 + 0.0494i)14-s + (1.63 + 0.689i)15-s + (0.661 − 0.750i)16-s − 0.675i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0569i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.998 + 0.0569i$
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: \((2,\ 48,\ (\ :1),\ -0.998 + 0.0569i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0108703 - 0.381292i\)
\(L(\frac12)\) \(\approx\) \(0.0108703 - 0.381292i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.420 + 1.95i)T \)
3 \( 1 + (2.77 - 1.13i)T \)
good5 \( 1 + (6.28 + 6.28i)T + 25iT^{2} \)
7 \( 1 + 1.64iT - 49T^{2} \)
11 \( 1 + (-4.75 - 4.75i)T + 121iT^{2} \)
13 \( 1 + (9.35 + 9.35i)T + 169iT^{2} \)
17 \( 1 + 11.4iT - 289T^{2} \)
19 \( 1 + (8.58 + 8.58i)T + 361iT^{2} \)
23 \( 1 + 16.2T + 529T^{2} \)
29 \( 1 + (-10.7 + 10.7i)T - 841iT^{2} \)
31 \( 1 - 6.35T + 961T^{2} \)
37 \( 1 + (-27.2 + 27.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 1.98T + 1.68e3T^{2} \)
43 \( 1 + (19.4 - 19.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 74.9iT - 2.20e3T^{2} \)
53 \( 1 + (4.00 + 4.00i)T + 2.80e3iT^{2} \)
59 \( 1 + (-27.9 - 27.9i)T + 3.48e3iT^{2} \)
61 \( 1 + (-39.2 - 39.2i)T + 3.72e3iT^{2} \)
67 \( 1 + (68.6 + 68.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 40.6T + 5.04e3T^{2} \)
73 \( 1 + 59.0iT - 5.32e3T^{2} \)
79 \( 1 - 17.3T + 6.24e3T^{2} \)
83 \( 1 + (-75.1 + 75.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 78.8T + 7.92e3T^{2} \)
97 \( 1 + 38.8T + 9.40e3T^{2} \)
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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

−15.09399185297670360240655984257, −13.11322575830481081067508700618, −12.14557465258064673485803617207, −11.63246926978903964805384135338, −10.24569092567341203423363296218, −9.028915912621668403166972937246, −7.60912145342794600397523228995, −5.00628057703952830532706431236, −4.07396037133073444669557435040, −0.48565899256073123151709289078, 4.22827261061241541052792511477, 6.17518696605526415138512955089, 7.08046938714429968738109920332, 8.179094024055993005343941130097, 10.17034871615443472834136512792, 11.33184502389508926295802905803, 12.39164850117926826696362941947, 14.10078597118179179123715473724, 15.02464322789001937362383620960, 16.02520931282412596094856957392

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