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Properties

Label 2-384-16.5-c3-0-13
Degree $2$
Conductor $384$
Sign $0.127 + 0.991i$
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  + (2.12 − 2.12i)3-s + (−3.22 − 3.22i)5-s + 24.6i·7-s − 8.99i·9-s + (−23.7 − 23.7i)11-s + (18.6 − 18.6i)13-s − 13.6·15-s − 3.55·17-s + (109. − 109. i)19-s + (52.2 + 52.2i)21-s − 36.5i·23-s − 104. i·25-s + (−19.0 − 19.0i)27-s + (−68.8 + 68.8i)29-s + 306.·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.288 − 0.288i)5-s + 1.32i·7-s − 0.333i·9-s + (−0.651 − 0.651i)11-s + (0.398 − 0.398i)13-s − 0.235·15-s − 0.0506·17-s + (1.31 − 1.31i)19-s + (0.542 + 0.542i)21-s − 0.331i·23-s − 0.833i·25-s + (−0.136 − 0.136i)27-s + (−0.440 + 0.440i)29-s + 1.77·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.127 + 0.991i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.127 + 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.759855901\)
\(L(\frac12)\) \(\approx\) \(1.759855901\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.12 + 2.12i)T \)
good5 \( 1 + (3.22 + 3.22i)T + 125iT^{2} \)
7 \( 1 - 24.6iT - 343T^{2} \)
11 \( 1 + (23.7 + 23.7i)T + 1.33e3iT^{2} \)
13 \( 1 + (-18.6 + 18.6i)T - 2.19e3iT^{2} \)
17 \( 1 + 3.55T + 4.91e3T^{2} \)
19 \( 1 + (-109. + 109. i)T - 6.85e3iT^{2} \)
23 \( 1 + 36.5iT - 1.21e4T^{2} \)
29 \( 1 + (68.8 - 68.8i)T - 2.43e4iT^{2} \)
31 \( 1 - 306.T + 2.97e4T^{2} \)
37 \( 1 + (92.9 + 92.9i)T + 5.06e4iT^{2} \)
41 \( 1 + 385. iT - 6.89e4T^{2} \)
43 \( 1 + (150. + 150. i)T + 7.95e4iT^{2} \)
47 \( 1 + 114.T + 1.03e5T^{2} \)
53 \( 1 + (451. + 451. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-544. - 544. i)T + 2.05e5iT^{2} \)
61 \( 1 + (179. - 179. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-283. + 283. i)T - 3.00e5iT^{2} \)
71 \( 1 + 930. iT - 3.57e5T^{2} \)
73 \( 1 - 701. iT - 3.89e5T^{2} \)
79 \( 1 - 779.T + 4.93e5T^{2} \)
83 \( 1 + (-296. + 296. i)T - 5.71e5iT^{2} \)
89 \( 1 - 865. iT - 7.04e5T^{2} \)
97 \( 1 + 542.T + 9.12e5T^{2} \)
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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

−10.81557516212830197872242846602, −9.579335221504970325464397757668, −8.638280267113558183910512890273, −8.198232803619595102404312477070, −6.97030941539938766359115554606, −5.80966167406210391762550280664, −4.95520909008884940354271121464, −3.27436936123762855726246477636, −2.37359370762271952888801696594, −0.61097133496676883928125854999, 1.36694604192606452769334534561, 3.13063820017043956504591520381, 4.02618309830012066997210223669, 5.05271345350411278896654438459, 6.52677461776845140384986364212, 7.60604515861630099013268295707, 8.064267648863221351656131948629, 9.639415365648284880543179281582, 10.04973535238694551270064834853, 11.02429923038858425018515676507

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