Properties

Label 2-48-16.3-c4-0-12
Degree $2$
Conductor $48$
Sign $0.995 + 0.0982i$
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  + (3.79 + 1.25i)2-s + (3.67 − 3.67i)3-s + (12.8 + 9.53i)4-s + (27.4 − 27.4i)5-s + (18.5 − 9.34i)6-s − 74.9·7-s + (36.8 + 52.3i)8-s − 27i·9-s + (138. − 69.8i)10-s + (107. + 107. i)11-s + (82.2 − 12.1i)12-s + (36.5 + 36.5i)13-s + (−284. − 94.0i)14-s − 201. i·15-s + (74.2 + 244. i)16-s − 343.·17-s + ⋯
L(s)  = 1  + (0.949 + 0.313i)2-s + (0.408 − 0.408i)3-s + (0.803 + 0.595i)4-s + (1.09 − 1.09i)5-s + (0.515 − 0.259i)6-s − 1.53·7-s + (0.575 + 0.817i)8-s − 0.333i·9-s + (1.38 − 0.698i)10-s + (0.886 + 0.886i)11-s + (0.571 − 0.0846i)12-s + (0.216 + 0.216i)13-s + (−1.45 − 0.480i)14-s − 0.897i·15-s + (0.290 + 0.957i)16-s − 1.18·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0982i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 + 0.0982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.995 + 0.0982i$
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: \((2,\ 48,\ (\ :2),\ 0.995 + 0.0982i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.98671 - 0.147092i\)
\(L(\frac12)\) \(\approx\) \(2.98671 - 0.147092i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.79 - 1.25i)T \)
3 \( 1 + (-3.67 + 3.67i)T \)
good5 \( 1 + (-27.4 + 27.4i)T - 625iT^{2} \)
7 \( 1 + 74.9T + 2.40e3T^{2} \)
11 \( 1 + (-107. - 107. i)T + 1.46e4iT^{2} \)
13 \( 1 + (-36.5 - 36.5i)T + 2.85e4iT^{2} \)
17 \( 1 + 343.T + 8.35e4T^{2} \)
19 \( 1 + (102. - 102. i)T - 1.30e5iT^{2} \)
23 \( 1 + 436.T + 2.79e5T^{2} \)
29 \( 1 + (535. + 535. i)T + 7.07e5iT^{2} \)
31 \( 1 - 769. iT - 9.23e5T^{2} \)
37 \( 1 + (-1.69e3 + 1.69e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.69e3iT - 2.82e6T^{2} \)
43 \( 1 + (1.13e3 + 1.13e3i)T + 3.41e6iT^{2} \)
47 \( 1 + 979. iT - 4.87e6T^{2} \)
53 \( 1 + (-2.63e3 + 2.63e3i)T - 7.89e6iT^{2} \)
59 \( 1 + (-4.78e3 - 4.78e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (415. + 415. i)T + 1.38e7iT^{2} \)
67 \( 1 + (-62.1 + 62.1i)T - 2.01e7iT^{2} \)
71 \( 1 + 1.21e3T + 2.54e7T^{2} \)
73 \( 1 - 3.71e3iT - 2.83e7T^{2} \)
79 \( 1 + 9.37e3iT - 3.89e7T^{2} \)
83 \( 1 + (2.90e3 - 2.90e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 4.49e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.23e4T + 8.85e7T^{2} \)
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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

−14.67401557104043529047354487647, −13.37186538605896019049597331440, −13.03148615642957591498838901713, −12.01811539181386342225209020544, −9.841510071973127809546200206003, −8.815137042653121203612443536820, −6.85877826428708519609554493180, −5.93782613921887748301286530027, −4.11994543221557853437233458105, −2.07672162635362361345663986941, 2.55006108098987947864789408236, 3.70216510137155053153384537254, 6.02660680983604352284622915701, 6.67014045980801794072029219919, 9.324173440710981847723089790404, 10.24135982706171485701109647954, 11.29697228292041621528707292678, 13.09119101320204043608762127171, 13.69302233603059240591008014753, 14.68672724156946734421956110076

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