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Properties

Label 2-384-12.11-c3-0-13
Degree $2$
Conductor $384$
Sign $0.485 - 0.874i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.52 − 4.54i)3-s + 8.01i·5-s + 12.6i·7-s + (−14.2 − 22.9i)9-s − 37.7·11-s + 60.0·13-s + (36.3 + 20.2i)15-s + 37.0i·17-s + 127. i·19-s + (57.2 + 31.7i)21-s − 56.9·23-s + 60.8·25-s + (−140. + 7.09i)27-s + 220. i·29-s − 2.26i·31-s + ⋯
L(s)  = 1  + (0.485 − 0.874i)3-s + 0.716i·5-s + 0.680i·7-s + (−0.528 − 0.848i)9-s − 1.03·11-s + 1.28·13-s + (0.626 + 0.347i)15-s + 0.528i·17-s + 1.54i·19-s + (0.594 + 0.330i)21-s − 0.516·23-s + 0.486·25-s + (−0.998 + 0.0505i)27-s + 1.41i·29-s − 0.0130i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.485 - 0.874i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.485 - 0.874i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.768083329\)
\(L(\frac12)\) \(\approx\) \(1.768083329\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.52 + 4.54i)T \)
good5 \( 1 - 8.01iT - 125T^{2} \)
7 \( 1 - 12.6iT - 343T^{2} \)
11 \( 1 + 37.7T + 1.33e3T^{2} \)
13 \( 1 - 60.0T + 2.19e3T^{2} \)
17 \( 1 - 37.0iT - 4.91e3T^{2} \)
19 \( 1 - 127. iT - 6.85e3T^{2} \)
23 \( 1 + 56.9T + 1.21e4T^{2} \)
29 \( 1 - 220. iT - 2.43e4T^{2} \)
31 \( 1 + 2.26iT - 2.97e4T^{2} \)
37 \( 1 + 166.T + 5.06e4T^{2} \)
41 \( 1 - 154. iT - 6.89e4T^{2} \)
43 \( 1 - 53.4iT - 7.95e4T^{2} \)
47 \( 1 - 591.T + 1.03e5T^{2} \)
53 \( 1 - 538. iT - 1.48e5T^{2} \)
59 \( 1 + 586.T + 2.05e5T^{2} \)
61 \( 1 - 431.T + 2.26e5T^{2} \)
67 \( 1 + 175. iT - 3.00e5T^{2} \)
71 \( 1 - 29.5T + 3.57e5T^{2} \)
73 \( 1 - 937.T + 3.89e5T^{2} \)
79 \( 1 + 409. iT - 4.93e5T^{2} \)
83 \( 1 + 1.29e3T + 5.71e5T^{2} \)
89 \( 1 + 921. iT - 7.04e5T^{2} \)
97 \( 1 + 1.64e3T + 9.12e5T^{2} \)
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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

−10.99262526706946358944587671738, −10.32637377684665737926801295805, −8.944314349715416583228976093568, −8.263418244949274258176177653753, −7.40527833949433215020137638715, −6.28925376262043179038309252245, −5.61645082642706056192833863817, −3.68405403730489997367669122889, −2.71903700115933382803633972379, −1.49566555769491626095305606309, 0.57457056987600209402849644585, 2.49328935947762965690096843287, 3.80418364564227725805408989895, 4.71920631334342625245947661864, 5.63140383053465349498956535113, 7.14964927338936973713583808041, 8.246184876868177211379563730752, 8.872212565047351980714333752967, 9.838613487999027315815583998413, 10.71866194950422581156682120654

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