Properties

Label 2-48-12.11-c21-0-23
Degree $2$
Conductor $48$
Sign $0.673 + 0.738i$
Analytic cond. $134.149$
Root an. cond. $11.5822$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.10e4 − 9.74e4i)3-s + 2.82e7i·5-s + 1.02e8i·7-s + (−8.53e9 − 6.04e9i)9-s + 1.32e11·11-s − 9.58e11·13-s + (2.75e12 + 8.76e11i)15-s + 2.11e12i·17-s − 5.68e12i·19-s + (1.00e13 + 3.19e12i)21-s − 9.10e13·23-s − 3.22e14·25-s + (−8.53e14 + 6.44e14i)27-s − 3.40e15i·29-s − 2.15e15i·31-s + ⋯
L(s)  = 1  + (0.303 − 0.952i)3-s + 1.29i·5-s + 0.137i·7-s + (−0.816 − 0.577i)9-s + 1.54·11-s − 1.92·13-s + (1.23 + 0.392i)15-s + 0.254i·17-s − 0.212i·19-s + (0.131 + 0.0417i)21-s − 0.458·23-s − 0.676·25-s + (−0.797 + 0.602i)27-s − 1.50i·29-s − 0.471i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.738i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.673 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.673 + 0.738i$
Analytic conductor: \(134.149\)
Root analytic conductor: \(11.5822\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :21/2),\ 0.673 + 0.738i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.051341696\)
\(L(\frac12)\) \(\approx\) \(2.051341696\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-3.10e4 + 9.74e4i)T \)
good5 \( 1 - 2.82e7iT - 4.76e14T^{2} \)
7 \( 1 - 1.02e8iT - 5.58e17T^{2} \)
11 \( 1 - 1.32e11T + 7.40e21T^{2} \)
13 \( 1 + 9.58e11T + 2.47e23T^{2} \)
17 \( 1 - 2.11e12iT - 6.90e25T^{2} \)
19 \( 1 + 5.68e12iT - 7.14e26T^{2} \)
23 \( 1 + 9.10e13T + 3.94e28T^{2} \)
29 \( 1 + 3.40e15iT - 5.13e30T^{2} \)
31 \( 1 + 2.15e15iT - 2.08e31T^{2} \)
37 \( 1 + 2.84e16T + 8.55e32T^{2} \)
41 \( 1 - 2.99e16iT - 7.38e33T^{2} \)
43 \( 1 + 6.56e16iT - 2.00e34T^{2} \)
47 \( 1 - 6.12e17T + 1.30e35T^{2} \)
53 \( 1 - 1.81e18iT - 1.62e36T^{2} \)
59 \( 1 - 4.67e18T + 1.54e37T^{2} \)
61 \( 1 - 6.14e18T + 3.10e37T^{2} \)
67 \( 1 - 1.06e19iT - 2.22e38T^{2} \)
71 \( 1 + 2.58e19T + 7.52e38T^{2} \)
73 \( 1 + 1.97e19T + 1.34e39T^{2} \)
79 \( 1 + 1.51e20iT - 7.08e39T^{2} \)
83 \( 1 - 1.89e20T + 1.99e40T^{2} \)
89 \( 1 - 2.33e20iT - 8.65e40T^{2} \)
97 \( 1 - 4.64e20T + 5.27e41T^{2} \)
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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

−11.65074099666962021911395351970, −10.16355540986945712180428527230, −9.031805214902551600229499849116, −7.53072502642904980915773131744, −6.90052997637886873602638584932, −5.91426442033130811941436265491, −4.02266880597164504742772221403, −2.72666140934624804211960570265, −2.01474391722111379335557940914, −0.52025155976046779624047414931, 0.76602067353757326257219987909, 2.09642423814599634725175904333, 3.63308206313989874065868512137, 4.62259623471215877462285361377, 5.37076811289125214635596850440, 7.13453952274186370157206620195, 8.617863400541739287861952882530, 9.279013358549163866278698047113, 10.20500642498547520508336639230, 11.77609973407352939264881677659

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