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Properties

Label 2-48-3.2-c6-0-2
Degree $2$
Conductor $48$
Sign $-0.814 - 0.579i$
Analytic cond. $11.0425$
Root an. cond. $3.32304$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (22.0 + 15.6i)3-s + 161. i·5-s − 562.·7-s + (239. + 688. i)9-s − 1.45e3i·11-s − 1.09e3·13-s + (−2.52e3 + 3.55e3i)15-s + 4.05e3i·17-s − 1.57e3·19-s + (−1.23e4 − 8.79e3i)21-s + 1.15e4i·23-s − 1.04e4·25-s + (−5.51e3 + 1.88e4i)27-s + 3.72e4i·29-s + 2.34e3·31-s + ⋯
L(s)  = 1  + (0.814 + 0.579i)3-s + 1.29i·5-s − 1.63·7-s + (0.328 + 0.944i)9-s − 1.09i·11-s − 0.499·13-s + (−0.748 + 1.05i)15-s + 0.825i·17-s − 0.229·19-s + (−1.33 − 0.949i)21-s + 0.950i·23-s − 0.667·25-s + (−0.280 + 0.959i)27-s + 1.52i·29-s + 0.0786·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.814 - 0.579i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ -0.814 - 0.579i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.427105 + 1.33726i\)
\(L(\frac12)\) \(\approx\) \(0.427105 + 1.33726i\)
\(L(4)\) 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 \( 1 + (-22.0 - 15.6i)T \)
good5 \( 1 - 161. iT - 1.56e4T^{2} \)
7 \( 1 + 562.T + 1.17e5T^{2} \)
11 \( 1 + 1.45e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.09e3T + 4.82e6T^{2} \)
17 \( 1 - 4.05e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.57e3T + 4.70e7T^{2} \)
23 \( 1 - 1.15e4iT - 1.48e8T^{2} \)
29 \( 1 - 3.72e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.34e3T + 8.87e8T^{2} \)
37 \( 1 - 6.80e4T + 2.56e9T^{2} \)
41 \( 1 + 3.60e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.01e5T + 6.32e9T^{2} \)
47 \( 1 - 5.03e3iT - 1.07e10T^{2} \)
53 \( 1 + 1.09e5iT - 2.21e10T^{2} \)
59 \( 1 - 2.50e5iT - 4.21e10T^{2} \)
61 \( 1 + 3.18e5T + 5.15e10T^{2} \)
67 \( 1 + 2.26e5T + 9.04e10T^{2} \)
71 \( 1 - 2.35e5iT - 1.28e11T^{2} \)
73 \( 1 - 6.94e5T + 1.51e11T^{2} \)
79 \( 1 + 2.05e5T + 2.43e11T^{2} \)
83 \( 1 + 1.26e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.03e6iT - 4.96e11T^{2} \)
97 \( 1 - 3.17e5T + 8.32e11T^{2} \)
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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

−14.85517223485433054387017654298, −13.89062463041542764517718807114, −12.83125179407270463342080511579, −10.94807263062548320603236304414, −10.09467422447685816769800980014, −8.993525857772256015263976021416, −7.35224627573673099693310376564, −6.06607789808751555199274179005, −3.61576263944256281166057072559, −2.81505493946119503805598678888, 0.55472109528060079629770523674, 2.54420280282852575451194580206, 4.38646535495858543981012351283, 6.40164570611060507035531503177, 7.72655488189807329288142256754, 9.215725449833676232110205835569, 9.743168576178480508561155973663, 12.27460203469801152262412364015, 12.71585990117880398262916100023, 13.64741247937605761296885423110

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