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Properties

Label 2-384-24.5-c2-0-25
Degree $2$
Conductor $384$
Sign $-0.543 + 0.839i$
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  + (0.628 + 2.93i)3-s − 6.51·5-s + 7.64·7-s + (−8.21 + 3.68i)9-s − 17.8·11-s − 14.4i·13-s + (−4.09 − 19.1i)15-s + 18.6i·17-s − 12.9i·19-s + (4.80 + 22.4i)21-s − 28.1i·23-s + 17.4·25-s + (−15.9 − 21.7i)27-s − 3.08·29-s − 32.1·31-s + ⋯
L(s)  = 1  + (0.209 + 0.977i)3-s − 1.30·5-s + 1.09·7-s + (−0.912 + 0.409i)9-s − 1.62·11-s − 1.10i·13-s + (−0.272 − 1.27i)15-s + 1.09i·17-s − 0.682i·19-s + (0.228 + 1.06i)21-s − 1.22i·23-s + 0.696·25-s + (−0.591 − 0.806i)27-s − 0.106·29-s − 1.03·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.543 + 0.839i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.543 + 0.839i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0695266 - 0.127827i\)
\(L(\frac12)\) \(\approx\) \(0.0695266 - 0.127827i\)
\(L(2)\) 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 \( 1 + (-0.628 - 2.93i)T \)
good5 \( 1 + 6.51T + 25T^{2} \)
7 \( 1 - 7.64T + 49T^{2} \)
11 \( 1 + 17.8T + 121T^{2} \)
13 \( 1 + 14.4iT - 169T^{2} \)
17 \( 1 - 18.6iT - 289T^{2} \)
19 \( 1 + 12.9iT - 361T^{2} \)
23 \( 1 + 28.1iT - 529T^{2} \)
29 \( 1 + 3.08T + 841T^{2} \)
31 \( 1 + 32.1T + 961T^{2} \)
37 \( 1 + 1.57iT - 1.36e3T^{2} \)
41 \( 1 + 18.1iT - 1.68e3T^{2} \)
43 \( 1 - 22.2iT - 1.84e3T^{2} \)
47 \( 1 + 86.4iT - 2.20e3T^{2} \)
53 \( 1 + 25.7T + 2.80e3T^{2} \)
59 \( 1 - 11.3T + 3.48e3T^{2} \)
61 \( 1 - 91.2iT - 3.72e3T^{2} \)
67 \( 1 + 17.6iT - 4.48e3T^{2} \)
71 \( 1 - 58.3iT - 5.04e3T^{2} \)
73 \( 1 + 104.T + 5.32e3T^{2} \)
79 \( 1 + 123.T + 6.24e3T^{2} \)
83 \( 1 - 81.7T + 6.88e3T^{2} \)
89 \( 1 - 142. iT - 7.92e3T^{2} \)
97 \( 1 + 58.1T + 9.40e3T^{2} \)
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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

−10.74363411320732349452813710023, −10.26926583136193674212465560230, −8.607173334485875017427114317579, −8.195930589017214024084488510335, −7.45778473235947603885537091168, −5.55690845649989367381971090938, −4.79092239198237178121646477714, −3.82100530433719542930081728725, −2.61354046540234182095534932660, −0.06037910283505228001836728839, 1.73402319797150705958558248303, 3.14361669089042621853338693961, 4.54559253426446515262640236438, 5.60022561089491345708317079559, 7.18079672042095012412415031859, 7.69822137049605082329265400470, 8.257505619712841729124419907787, 9.400453893045795137864757859410, 10.99909111692403310656691083197, 11.46902725388723897136771032287

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