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Properties

Label 2-384-48.5-c2-0-4
Degree $2$
Conductor $384$
Sign $0.264 - 0.964i$
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  + (−1.50 − 2.59i)3-s + (2.59 − 2.59i)5-s + 7.30i·7-s + (−4.47 + 7.81i)9-s + (−11.3 + 11.3i)11-s + (0.746 − 0.746i)13-s + (−10.6 − 2.83i)15-s − 6.67i·17-s + (−22.1 + 22.1i)19-s + (18.9 − 10.9i)21-s + 21.4·23-s + 11.4i·25-s + (26.9 − 0.153i)27-s + (−1.54 − 1.54i)29-s − 14.6·31-s + ⋯
L(s)  = 1  + (−0.501 − 0.865i)3-s + (0.519 − 0.519i)5-s + 1.04i·7-s + (−0.496 + 0.867i)9-s + (−1.02 + 1.02i)11-s + (0.0574 − 0.0574i)13-s + (−0.710 − 0.188i)15-s − 0.392i·17-s + (−1.16 + 1.16i)19-s + (0.902 − 0.523i)21-s + 0.932·23-s + 0.459i·25-s + (0.999 − 0.00567i)27-s + (−0.0531 − 0.0531i)29-s − 0.471·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.264 - 0.964i$
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: \((2,\ 384,\ (\ :1),\ 0.264 - 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.724761 + 0.552628i\)
\(L(\frac12)\) \(\approx\) \(0.724761 + 0.552628i\)
\(L(2)\) 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 \( 1 + (1.50 + 2.59i)T \)
good5 \( 1 + (-2.59 + 2.59i)T - 25iT^{2} \)
7 \( 1 - 7.30iT - 49T^{2} \)
11 \( 1 + (11.3 - 11.3i)T - 121iT^{2} \)
13 \( 1 + (-0.746 + 0.746i)T - 169iT^{2} \)
17 \( 1 + 6.67iT - 289T^{2} \)
19 \( 1 + (22.1 - 22.1i)T - 361iT^{2} \)
23 \( 1 - 21.4T + 529T^{2} \)
29 \( 1 + (1.54 + 1.54i)T + 841iT^{2} \)
31 \( 1 + 14.6T + 961T^{2} \)
37 \( 1 + (-50.1 - 50.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 15.0T + 1.68e3T^{2} \)
43 \( 1 + (26.3 + 26.3i)T + 1.84e3iT^{2} \)
47 \( 1 - 36.6iT - 2.20e3T^{2} \)
53 \( 1 + (50.9 - 50.9i)T - 2.80e3iT^{2} \)
59 \( 1 + (-12.1 + 12.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (-27.5 + 27.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (-4.84 + 4.84i)T - 4.48e3iT^{2} \)
71 \( 1 + 74.9T + 5.04e3T^{2} \)
73 \( 1 + 3.47iT - 5.32e3T^{2} \)
79 \( 1 - 103.T + 6.24e3T^{2} \)
83 \( 1 + (-31.7 - 31.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 78.2T + 7.92e3T^{2} \)
97 \( 1 + 61.5T + 9.40e3T^{2} \)
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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

−11.46402958821366520237224094630, −10.45690121526205040879309368667, −9.453798535831809111412765933239, −8.435125888725554333140018798184, −7.60174796100280828993882043302, −6.43453566428692596867556788269, −5.53339020469945186650049453746, −4.81081271305496760070272081884, −2.63281106343645924156443725709, −1.62783402359090038259092054609, 0.41774424184818042077058480175, 2.73106762737742875699580038913, 3.93074692308462116824785062620, 5.02591854134316071306770598670, 6.08619165116066860385822804762, 6.93631483845130648109754207217, 8.262260965419369078313621699141, 9.273315168373818139066910531873, 10.36392796332876447062823728850, 10.76897074299536754009736409945

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