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Properties

Label 2-48-16.11-c4-0-7
Degree $2$
Conductor $48$
Sign $0.958 - 0.285i$
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  + (−1.62 + 3.65i)2-s + (−3.67 − 3.67i)3-s + (−10.7 − 11.8i)4-s + (−2.10 − 2.10i)5-s + (19.4 − 7.45i)6-s + 84.8·7-s + (60.8 − 19.8i)8-s + 27i·9-s + (11.1 − 4.26i)10-s + (−57.8 + 57.8i)11-s + (−4.30 + 83.0i)12-s + (192. − 192. i)13-s + (−137. + 310. i)14-s + 15.4i·15-s + (−26.4 + 254. i)16-s + 507.·17-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.408 − 0.408i)3-s + (−0.669 − 0.742i)4-s + (−0.0841 − 0.0841i)5-s + (0.538 − 0.207i)6-s + 1.73·7-s + (0.950 − 0.309i)8-s + 0.333i·9-s + (0.111 − 0.0426i)10-s + (−0.478 + 0.478i)11-s + (−0.0298 + 0.576i)12-s + (1.13 − 1.13i)13-s + (−0.703 + 1.58i)14-s + 0.0686i·15-s + (−0.103 + 0.994i)16-s + 1.75·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.958 - 0.285i$
Analytic conductor: \(4.96175\)
Root analytic conductor: \(2.22750\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :2),\ 0.958 - 0.285i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.19148 + 0.173460i\)
\(L(\frac12)\) \(\approx\) \(1.19148 + 0.173460i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.62 - 3.65i)T \)
3 \( 1 + (3.67 + 3.67i)T \)
good5 \( 1 + (2.10 + 2.10i)T + 625iT^{2} \)
7 \( 1 - 84.8T + 2.40e3T^{2} \)
11 \( 1 + (57.8 - 57.8i)T - 1.46e4iT^{2} \)
13 \( 1 + (-192. + 192. i)T - 2.85e4iT^{2} \)
17 \( 1 - 507.T + 8.35e4T^{2} \)
19 \( 1 + (126. + 126. i)T + 1.30e5iT^{2} \)
23 \( 1 + 173.T + 2.79e5T^{2} \)
29 \( 1 + (64.7 - 64.7i)T - 7.07e5iT^{2} \)
31 \( 1 + 366. iT - 9.23e5T^{2} \)
37 \( 1 + (-801. - 801. i)T + 1.87e6iT^{2} \)
41 \( 1 - 461. iT - 2.82e6T^{2} \)
43 \( 1 + (1.14e3 - 1.14e3i)T - 3.41e6iT^{2} \)
47 \( 1 - 4.09e3iT - 4.87e6T^{2} \)
53 \( 1 + (2.54e3 + 2.54e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-959. + 959. i)T - 1.21e7iT^{2} \)
61 \( 1 + (162. - 162. i)T - 1.38e7iT^{2} \)
67 \( 1 + (771. + 771. i)T + 2.01e7iT^{2} \)
71 \( 1 + 2.06e3T + 2.54e7T^{2} \)
73 \( 1 + 463. iT - 2.83e7T^{2} \)
79 \( 1 - 8.43e3iT - 3.89e7T^{2} \)
83 \( 1 + (3.10e3 + 3.10e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 7.66e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.63e4T + 8.85e7T^{2} \)
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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

−15.00093405337928808629749603271, −14.11459806516311403535564872962, −12.79931442473778153588164473375, −11.26975889852367319420602467887, −10.18360924687971113510761292206, −8.205465591186543104223494256351, −7.78718949056925925600383951112, −5.93075221726187382988688678997, −4.80254516463027911975774976188, −1.18076393640357654506399985421, 1.46741881880643972303284209367, 3.84918256698835893793471397880, 5.32908376152001975564898990325, 7.78858487966555951696017815058, 8.842219114232396950420172213415, 10.40791153086570965975302145514, 11.27229466653694056564163712137, 12.00615577123908373932094955426, 13.68289351885294347517360313573, 14.65304657684499202739665804147

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