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Properties

Label 2-384-16.13-c3-0-8
Degree $2$
Conductor $384$
Sign $-0.671 - 0.740i$
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 + (8.83 − 8.83i)5-s + 29.4i·7-s + 8.99i·9-s + (−44.6 + 44.6i)11-s + (−6.83 − 6.83i)13-s + 37.4·15-s − 56.8·17-s + (−91.0 − 91.0i)19-s + (−62.5 + 62.5i)21-s + 96.8i·23-s − 31.0i·25-s + (−19.0 + 19.0i)27-s + (59.2 + 59.2i)29-s + 103.·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.790 − 0.790i)5-s + 1.59i·7-s + 0.333i·9-s + (−1.22 + 1.22i)11-s + (−0.145 − 0.145i)13-s + 0.645·15-s − 0.810·17-s + (−1.09 − 1.09i)19-s + (−0.649 + 0.649i)21-s + 0.877i·23-s − 0.248i·25-s + (−0.136 + 0.136i)27-s + (0.379 + 0.379i)29-s + 0.597·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.671 - 0.740i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.671 - 0.740i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.512399160\)
\(L(\frac12)\) \(\approx\) \(1.512399160\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.83 + 8.83i)T - 125iT^{2} \)
7 \( 1 - 29.4iT - 343T^{2} \)
11 \( 1 + (44.6 - 44.6i)T - 1.33e3iT^{2} \)
13 \( 1 + (6.83 + 6.83i)T + 2.19e3iT^{2} \)
17 \( 1 + 56.8T + 4.91e3T^{2} \)
19 \( 1 + (91.0 + 91.0i)T + 6.85e3iT^{2} \)
23 \( 1 - 96.8iT - 1.21e4T^{2} \)
29 \( 1 + (-59.2 - 59.2i)T + 2.43e4iT^{2} \)
31 \( 1 - 103.T + 2.97e4T^{2} \)
37 \( 1 + (-79.5 + 79.5i)T - 5.06e4iT^{2} \)
41 \( 1 - 105. iT - 6.89e4T^{2} \)
43 \( 1 + (39.9 - 39.9i)T - 7.95e4iT^{2} \)
47 \( 1 - 9.34T + 1.03e5T^{2} \)
53 \( 1 + (245. - 245. i)T - 1.48e5iT^{2} \)
59 \( 1 + (345. - 345. i)T - 2.05e5iT^{2} \)
61 \( 1 + (370. + 370. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-595. - 595. i)T + 3.00e5iT^{2} \)
71 \( 1 + 493. iT - 3.57e5T^{2} \)
73 \( 1 + 33.2iT - 3.89e5T^{2} \)
79 \( 1 - 552.T + 4.93e5T^{2} \)
83 \( 1 + (-18.5 - 18.5i)T + 5.71e5iT^{2} \)
89 \( 1 - 934. iT - 7.04e5T^{2} \)
97 \( 1 - 1.42e3T + 9.12e5T^{2} \)
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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

−11.18174598043595860980087008803, −10.10644717625308856704965363640, −9.253805420983079435205578527246, −8.807009380791109065164114901652, −7.74012221603054183470849975341, −6.30256496035990104891922886846, −5.17768644811127585198739338090, −4.70262248530654974013673184860, −2.67603876362144089482230346656, −2.00613265572830040485755960036, 0.44151624102957382797499422560, 2.12133537909166934758092912223, 3.22926953532881899624707971866, 4.47672795669868327156137454084, 6.09326698001402026367876598999, 6.69488361452528856625835714750, 7.80366179822452519605832223570, 8.506610777484500099334483944285, 9.993524781109010520223249212042, 10.51340150593379283054623525833

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