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Properties

Label 2-384-48.5-c2-0-24
Degree $2$
Conductor $384$
Sign $-0.989 + 0.142i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.75 − 1.18i)3-s + (−0.00985 + 0.00985i)5-s − 6.42i·7-s + (6.19 + 6.53i)9-s + (9.07 − 9.07i)11-s + (−12.6 + 12.6i)13-s + (0.0388 − 0.0154i)15-s − 19.0i·17-s + (2.07 − 2.07i)19-s + (−7.61 + 17.7i)21-s − 19.5·23-s + 24.9i·25-s + (−9.32 − 25.3i)27-s + (−11.1 − 11.1i)29-s − 59.9·31-s + ⋯
L(s)  = 1  + (−0.918 − 0.395i)3-s + (−0.00197 + 0.00197i)5-s − 0.917i·7-s + (0.687 + 0.725i)9-s + (0.824 − 0.824i)11-s + (−0.969 + 0.969i)13-s + (0.00259 − 0.00103i)15-s − 1.11i·17-s + (0.109 − 0.109i)19-s + (−0.362 + 0.842i)21-s − 0.850·23-s + 0.999i·25-s + (−0.345 − 0.938i)27-s + (−0.385 − 0.385i)29-s − 1.93·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.989 + 0.142i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.989 + 0.142i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0330571 - 0.460211i\)
\(L(\frac12)\) \(\approx\) \(0.0330571 - 0.460211i\)
\(L(2)\) 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 \( 1 + (2.75 + 1.18i)T \)
good5 \( 1 + (0.00985 - 0.00985i)T - 25iT^{2} \)
7 \( 1 + 6.42iT - 49T^{2} \)
11 \( 1 + (-9.07 + 9.07i)T - 121iT^{2} \)
13 \( 1 + (12.6 - 12.6i)T - 169iT^{2} \)
17 \( 1 + 19.0iT - 289T^{2} \)
19 \( 1 + (-2.07 + 2.07i)T - 361iT^{2} \)
23 \( 1 + 19.5T + 529T^{2} \)
29 \( 1 + (11.1 + 11.1i)T + 841iT^{2} \)
31 \( 1 + 59.9T + 961T^{2} \)
37 \( 1 + (9.32 + 9.32i)T + 1.36e3iT^{2} \)
41 \( 1 + 47.2T + 1.68e3T^{2} \)
43 \( 1 + (24.1 + 24.1i)T + 1.84e3iT^{2} \)
47 \( 1 + 6.29iT - 2.20e3T^{2} \)
53 \( 1 + (-20.6 + 20.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (60.3 - 60.3i)T - 3.48e3iT^{2} \)
61 \( 1 + (48.0 - 48.0i)T - 3.72e3iT^{2} \)
67 \( 1 + (-23.7 + 23.7i)T - 4.48e3iT^{2} \)
71 \( 1 + 13.5T + 5.04e3T^{2} \)
73 \( 1 + 31.4iT - 5.32e3T^{2} \)
79 \( 1 - 47.4T + 6.24e3T^{2} \)
83 \( 1 + (70.3 + 70.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 95.1T + 7.92e3T^{2} \)
97 \( 1 - 61.6T + 9.40e3T^{2} \)
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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

−10.89701219144386617896316475219, −9.868866777642624500656055861911, −8.980274300856112284961684781914, −7.38826732438167101640215522402, −7.07503928213106243801921410241, −5.88645404061158310783917715757, −4.82151069606826525740088822969, −3.70890140630214096182495098312, −1.74189405630350123931959780258, −0.22463193524008575593133188587, 1.87666960785282079337315279073, 3.65030555840922234537548224800, 4.85011685446730536471067676195, 5.72880215769516624421523794141, 6.61861344228686852709368041838, 7.78066736549091874935404279339, 9.006963660925611678900548850080, 9.879871086712481200845152860497, 10.56219291262868328439088682091, 11.68296620526771563365189759469

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