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Properties

Label 2-384-8.5-c9-0-30
Degree $2$
Conductor $384$
Sign $0.707 - 0.707i$
Analytic cond. $197.773$
Root an. cond. $14.0632$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 81i·3-s − 309. i·5-s − 5.27e3·7-s − 6.56e3·9-s + 6.70e4i·11-s − 1.45e5i·13-s + 2.50e4·15-s + 2.80e5·17-s + 5.61e5i·19-s − 4.27e5i·21-s − 1.25e6·23-s + 1.85e6·25-s − 5.31e5i·27-s − 3.22e6i·29-s − 4.89e6·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.221i·5-s − 0.829·7-s − 0.333·9-s + 1.38i·11-s − 1.41i·13-s + 0.127·15-s + 0.815·17-s + 0.989i·19-s − 0.479i·21-s − 0.933·23-s + 0.950·25-s − 0.192i·27-s − 0.846i·29-s − 0.952·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(197.773\)
Root analytic conductor: \(14.0632\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :9/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.528483185\)
\(L(\frac12)\) \(\approx\) \(1.528483185\)
\(L(\frac{11}{2})\) 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 \( 1 - 81iT \)
good5 \( 1 + 309. iT - 1.95e6T^{2} \)
7 \( 1 + 5.27e3T + 4.03e7T^{2} \)
11 \( 1 - 6.70e4iT - 2.35e9T^{2} \)
13 \( 1 + 1.45e5iT - 1.06e10T^{2} \)
17 \( 1 - 2.80e5T + 1.18e11T^{2} \)
19 \( 1 - 5.61e5iT - 3.22e11T^{2} \)
23 \( 1 + 1.25e6T + 1.80e12T^{2} \)
29 \( 1 + 3.22e6iT - 1.45e13T^{2} \)
31 \( 1 + 4.89e6T + 2.64e13T^{2} \)
37 \( 1 + 1.02e7iT - 1.29e14T^{2} \)
41 \( 1 + 4.07e6T + 3.27e14T^{2} \)
43 \( 1 + 2.29e7iT - 5.02e14T^{2} \)
47 \( 1 + 9.57e6T + 1.11e15T^{2} \)
53 \( 1 + 5.95e7iT - 3.29e15T^{2} \)
59 \( 1 + 8.97e6iT - 8.66e15T^{2} \)
61 \( 1 - 1.82e8iT - 1.16e16T^{2} \)
67 \( 1 - 2.19e8iT - 2.72e16T^{2} \)
71 \( 1 + 2.11e8T + 4.58e16T^{2} \)
73 \( 1 - 4.05e8T + 5.88e16T^{2} \)
79 \( 1 + 1.72e8T + 1.19e17T^{2} \)
83 \( 1 + 7.57e8iT - 1.86e17T^{2} \)
89 \( 1 - 9.23e8T + 3.50e17T^{2} \)
97 \( 1 - 7.91e8T + 7.60e17T^{2} \)
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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

−10.20336859307233405661352696359, −9.207371181599317701115858452701, −8.080385230461061599283473038801, −7.24947145165000175612399761656, −5.97275160333542012813435581442, −5.22514821179547824967284576966, −4.04005044555187710650004520565, −3.20827971601924027992416885942, −1.99744210969398567531789371560, −0.56714863565169086120186923077, 0.46945855671240424512715283483, 1.56039556675661519150332882521, 2.85474765165127967625550943218, 3.59281142468982617666731111902, 5.02495117181384740677972550571, 6.26367514037240939159428377205, 6.68654476330085459648783626091, 7.81396045552184231015133987754, 8.847988637446772637724808423723, 9.527674737820811590844303226415

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