Properties

Label 2-384-8.3-c2-0-15
Degree $2$
Conductor $384$
Sign $-0.707 + 0.707i$
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  + 1.73·3-s − 8.89i·5-s − 2.82i·7-s + 2.99·9-s − 18.2·11-s − 5.79i·13-s − 15.4i·15-s − 21.5·17-s + 18.2·19-s − 4.89i·21-s + 33.3i·23-s − 54.1·25-s + 5.19·27-s − 4.49i·29-s − 2.25i·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.77i·5-s − 0.404i·7-s + 0.333·9-s − 1.65·11-s − 0.445i·13-s − 1.02i·15-s − 1.27·17-s + 0.960·19-s − 0.233i·21-s + 1.45i·23-s − 2.16·25-s + 0.192·27-s − 0.154i·29-s − 0.0728i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.543597 - 1.31236i\)
\(L(\frac12)\) \(\approx\) \(0.543597 - 1.31236i\)
\(L(2)\) 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 \( 1 - 1.73T \)
good5 \( 1 + 8.89iT - 25T^{2} \)
7 \( 1 + 2.82iT - 49T^{2} \)
11 \( 1 + 18.2T + 121T^{2} \)
13 \( 1 + 5.79iT - 169T^{2} \)
17 \( 1 + 21.5T + 289T^{2} \)
19 \( 1 - 18.2T + 361T^{2} \)
23 \( 1 - 33.3iT - 529T^{2} \)
29 \( 1 + 4.49iT - 841T^{2} \)
31 \( 1 + 2.25iT - 961T^{2} \)
37 \( 1 + 43.1iT - 1.36e3T^{2} \)
41 \( 1 + 1.59T + 1.68e3T^{2} \)
43 \( 1 - 63.4T + 1.84e3T^{2} \)
47 \( 1 + 72.3iT - 2.20e3T^{2} \)
53 \( 1 + 70.2iT - 2.80e3T^{2} \)
59 \( 1 - 34.6T + 3.48e3T^{2} \)
61 \( 1 + 63.5iT - 3.72e3T^{2} \)
67 \( 1 + 3.24T + 4.48e3T^{2} \)
71 \( 1 + 68.4iT - 5.04e3T^{2} \)
73 \( 1 + 10T + 5.32e3T^{2} \)
79 \( 1 + 35.0iT - 6.24e3T^{2} \)
83 \( 1 - 42.2T + 6.88e3T^{2} \)
89 \( 1 - 5.19T + 7.92e3T^{2} \)
97 \( 1 + 26.8T + 9.40e3T^{2} \)
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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

−10.68063862225455790121470763464, −9.650131052228941980580741284767, −8.925952750557589324188453998116, −8.030163122758195829753593191908, −7.40543289888976544612656408124, −5.56453226194417313722642746541, −4.93335492315791671880524551106, −3.75019784939746879223032331572, −2.13647827428484131105236094576, −0.55453423134577743180133730555, 2.48245251814859696055176850910, 2.85337990190841641218506003606, 4.40148409664798519766239810780, 5.87228877023309869152501390125, 6.90201880219013170882823205398, 7.61409454282058622358729147779, 8.640933342173182980023484772423, 9.792117506047038547521619084379, 10.62912705965786827211784867521, 11.14542889455126646091982571868

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