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Properties

Label 2-48-3.2-c26-0-5
Degree $2$
Conductor $48$
Sign $0.243 - 0.969i$
Analytic cond. $205.580$
Root an. cond. $14.3380$
Motivic weight $26$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.88e5 − 1.54e6i)3-s + 1.99e8i·5-s + 4.03e9·7-s + (−2.23e12 − 1.20e12i)9-s + 4.12e13i·11-s − 9.55e13·13-s + (3.08e14 + 7.74e13i)15-s − 5.48e15i·17-s + 2.57e16·19-s + (1.56e15 − 6.23e15i)21-s − 6.39e17i·23-s + 1.45e18·25-s + (−2.72e18 + 2.99e18i)27-s − 1.27e19i·29-s − 4.74e19·31-s + ⋯
L(s)  = 1  + (0.243 − 0.969i)3-s + 0.163i·5-s + 0.0416·7-s + (−0.881 − 0.472i)9-s + 1.19i·11-s − 0.315·13-s + (0.158 + 0.0397i)15-s − 0.553i·17-s + 0.612·19-s + (0.0101 − 0.0403i)21-s − 1.26i·23-s + 0.973·25-s + (−0.673 + 0.739i)27-s − 1.23i·29-s − 1.94·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.243 - 0.969i$
Analytic conductor: \(205.580\)
Root analytic conductor: \(14.3380\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :13),\ 0.243 - 0.969i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.8962426947\)
\(L(\frac12)\) \(\approx\) \(0.8962426947\)
\(L(14)\) 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 \( 1 + (-3.88e5 + 1.54e6i)T \)
good5 \( 1 - 1.99e8iT - 1.49e18T^{2} \)
7 \( 1 - 4.03e9T + 9.38e21T^{2} \)
11 \( 1 - 4.12e13iT - 1.19e27T^{2} \)
13 \( 1 + 9.55e13T + 9.17e28T^{2} \)
17 \( 1 + 5.48e15iT - 9.81e31T^{2} \)
19 \( 1 - 2.57e16T + 1.76e33T^{2} \)
23 \( 1 + 6.39e17iT - 2.54e35T^{2} \)
29 \( 1 + 1.27e19iT - 1.05e38T^{2} \)
31 \( 1 + 4.74e19T + 5.96e38T^{2} \)
37 \( 1 + 6.19e19T + 5.93e40T^{2} \)
41 \( 1 - 1.36e20iT - 8.55e41T^{2} \)
43 \( 1 - 2.89e21T + 2.95e42T^{2} \)
47 \( 1 - 1.44e21iT - 2.98e43T^{2} \)
53 \( 1 + 1.68e22iT - 6.77e44T^{2} \)
59 \( 1 - 1.69e23iT - 1.10e46T^{2} \)
61 \( 1 - 1.07e23T + 2.62e46T^{2} \)
67 \( 1 + 7.95e22T + 3.00e47T^{2} \)
71 \( 1 - 1.77e24iT - 1.35e48T^{2} \)
73 \( 1 + 2.59e24T + 2.79e48T^{2} \)
79 \( 1 + 5.88e24T + 2.17e49T^{2} \)
83 \( 1 - 7.37e24iT - 7.87e49T^{2} \)
89 \( 1 + 7.84e24iT - 4.83e50T^{2} \)
97 \( 1 + 7.65e25T + 4.52e51T^{2} \)
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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

−11.22424601075550780719267655252, −9.846090250573517562244945974226, −8.772343980828289267508791109050, −7.49691679136827893898521660723, −6.93140710287538536857674510603, −5.63614860321354875929831072173, −4.35366992981099277644200613306, −2.86287987496939010550441785634, −2.07482919289702022072734448514, −0.933043577740483969132706376996, 0.16896917294628039530735106349, 1.54201182034488602594181273061, 3.03487755510861801741763835321, 3.71676116832961440884431192344, 5.05339511979290584133322909982, 5.82052858406183172076794250841, 7.44816407477982027273149595660, 8.663665649020366574950861785588, 9.376581110349050047220587283499, 10.65209746930660441120645560872

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