Properties

Label 2-384-8.5-c9-0-59
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 − 1.42e3i·5-s − 6.38e3·7-s − 6.56e3·9-s − 4.07e3i·11-s − 1.54e5i·13-s + 1.15e5·15-s + 2.55e5·17-s − 1.01e6i·19-s − 5.16e5i·21-s + 1.56e6·23-s − 8.63e4·25-s − 5.31e5i·27-s + 7.26e6i·29-s + 6.11e5·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.02i·5-s − 1.00·7-s − 0.333·9-s − 0.0838i·11-s − 1.49i·13-s + 0.589·15-s + 0.742·17-s − 1.79i·19-s − 0.580i·21-s + 1.16·23-s − 0.0442·25-s − 0.192i·27-s + 1.90i·29-s + 0.119·31-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

\(L(5)\) \(\approx\) \(1.485990536\)
\(L(\frac12)\) \(\approx\) \(1.485990536\)
\(L(\frac{11}{2})\) 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 \( 1 + 1.42e3iT - 1.95e6T^{2} \)
7 \( 1 + 6.38e3T + 4.03e7T^{2} \)
11 \( 1 + 4.07e3iT - 2.35e9T^{2} \)
13 \( 1 + 1.54e5iT - 1.06e10T^{2} \)
17 \( 1 - 2.55e5T + 1.18e11T^{2} \)
19 \( 1 + 1.01e6iT - 3.22e11T^{2} \)
23 \( 1 - 1.56e6T + 1.80e12T^{2} \)
29 \( 1 - 7.26e6iT - 1.45e13T^{2} \)
31 \( 1 - 6.11e5T + 2.64e13T^{2} \)
37 \( 1 + 4.64e5iT - 1.29e14T^{2} \)
41 \( 1 - 2.04e7T + 3.27e14T^{2} \)
43 \( 1 + 2.95e7iT - 5.02e14T^{2} \)
47 \( 1 + 1.31e6T + 1.11e15T^{2} \)
53 \( 1 - 5.76e7iT - 3.29e15T^{2} \)
59 \( 1 + 6.35e7iT - 8.66e15T^{2} \)
61 \( 1 + 5.01e7iT - 1.16e16T^{2} \)
67 \( 1 + 2.90e8iT - 2.72e16T^{2} \)
71 \( 1 - 3.18e8T + 4.58e16T^{2} \)
73 \( 1 + 3.49e8T + 5.88e16T^{2} \)
79 \( 1 - 4.29e8T + 1.19e17T^{2} \)
83 \( 1 + 3.40e8iT - 1.86e17T^{2} \)
89 \( 1 - 5.24e8T + 3.50e17T^{2} \)
97 \( 1 + 9.61e8T + 7.60e17T^{2} \)
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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

−9.229897434145180577626971835137, −8.936336019810089074502849534554, −7.71373388421423520623104291602, −6.60649799762604873108224293121, −5.34152271405738872812215322598, −4.91052661089716271682764979677, −3.46500562820703275783027174727, −2.81277171185032955521382930704, −0.978508022618680261705702167875, −0.33899419126839113949954568720, 1.10001001593556781052250639235, 2.28495495700837836501449452144, 3.19364735505993219170434657755, 4.16539332779133446988305656669, 5.82881270201833419079192763737, 6.49797962027430175319741272068, 7.20336265983043882439607336527, 8.155479929969104159496331118143, 9.454424235740462807777684141783, 10.02652638071946398717600353792

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