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Properties

Label 2-384-24.5-c4-0-29
Degree $2$
Conductor $384$
Sign $0.515 - 0.856i$
Analytic cond. $39.6940$
Root an. cond. $6.30032$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.73 − 2.16i)3-s − 14.5·5-s + 70.6·7-s + (71.5 − 37.8i)9-s − 163.·11-s + 293. i·13-s + (−126. + 31.4i)15-s + 533. i·17-s + 314. i·19-s + (617. − 153. i)21-s − 547. i·23-s − 414.·25-s + (543. − 486. i)27-s + 493.·29-s + 1.04e3·31-s + ⋯
L(s)  = 1  + (0.970 − 0.240i)3-s − 0.580·5-s + 1.44·7-s + (0.883 − 0.467i)9-s − 1.34·11-s + 1.73i·13-s + (−0.563 + 0.139i)15-s + 1.84i·17-s + 0.870i·19-s + (1.39 − 0.347i)21-s − 1.03i·23-s − 0.663·25-s + (0.745 − 0.666i)27-s + 0.587·29-s + 1.08·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.515 - 0.856i$
Analytic conductor: \(39.6940\)
Root analytic conductor: \(6.30032\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :2),\ 0.515 - 0.856i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.640325125\)
\(L(\frac12)\) \(\approx\) \(2.640325125\)
\(L(3)\) 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 \( 1 + (-8.73 + 2.16i)T \)
good5 \( 1 + 14.5T + 625T^{2} \)
7 \( 1 - 70.6T + 2.40e3T^{2} \)
11 \( 1 + 163.T + 1.46e4T^{2} \)
13 \( 1 - 293. iT - 2.85e4T^{2} \)
17 \( 1 - 533. iT - 8.35e4T^{2} \)
19 \( 1 - 314. iT - 1.30e5T^{2} \)
23 \( 1 + 547. iT - 2.79e5T^{2} \)
29 \( 1 - 493.T + 7.07e5T^{2} \)
31 \( 1 - 1.04e3T + 9.23e5T^{2} \)
37 \( 1 + 550. iT - 1.87e6T^{2} \)
41 \( 1 + 339. iT - 2.82e6T^{2} \)
43 \( 1 - 2.58e3iT - 3.41e6T^{2} \)
47 \( 1 + 100. iT - 4.87e6T^{2} \)
53 \( 1 - 1.40e3T + 7.89e6T^{2} \)
59 \( 1 + 1.43e3T + 1.21e7T^{2} \)
61 \( 1 - 4.99e3iT - 1.38e7T^{2} \)
67 \( 1 - 168. iT - 2.01e7T^{2} \)
71 \( 1 - 5.26e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.24e3T + 2.83e7T^{2} \)
79 \( 1 - 7.42e3T + 3.89e7T^{2} \)
83 \( 1 - 116.T + 4.74e7T^{2} \)
89 \( 1 + 7.44e3iT - 6.27e7T^{2} \)
97 \( 1 + 8.27e3T + 8.85e7T^{2} \)
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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

−10.82350948220625370265178114782, −9.994485961583311642398127549920, −8.551923651968132476362997701762, −8.230362940241470114609713374029, −7.46805655732300266023791748369, −6.20885801559019772750111123820, −4.61456035975698843651695737086, −3.98833680554137273049515688912, −2.37180912297061904723211336175, −1.47696418037967423528813079006, 0.67086703906433564161267399068, 2.38373215486227259431584817804, 3.25162551119606568330059051549, 4.78781715057890828954716625398, 5.22759269939521478017462701720, 7.35748210667068393228811902191, 7.85454415299457607811659569074, 8.419385870751883416864219615768, 9.652990873901669483064636382470, 10.55967679161068384769089252885

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