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Properties

Label 2-384-8.5-c7-0-34
Degree $2$
Conductor $384$
Sign $1$
Analytic cond. $119.955$
Root an. cond. $10.9524$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27i·3-s + 484. i·5-s − 826.·7-s − 729·9-s − 4.55e3i·11-s − 3.07e3i·13-s − 1.30e4·15-s + 2.16e4·17-s + 5.88e4i·19-s − 2.23e4i·21-s − 1.86e4·23-s − 1.56e5·25-s − 1.96e4i·27-s − 2.10e5i·29-s − 9.87e4·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.73i·5-s − 0.911·7-s − 0.333·9-s − 1.03i·11-s − 0.387i·13-s − 1.00·15-s + 1.06·17-s + 1.96i·19-s − 0.526i·21-s − 0.319·23-s − 2.00·25-s − 0.192i·27-s − 1.60i·29-s − 0.595·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
Analytic conductor: \(119.955\)
Root analytic conductor: \(10.9524\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.205007190\)
\(L(\frac12)\) \(\approx\) \(1.205007190\)
\(L(\frac{9}{2})\) 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 \( 1 - 27iT \)
good5 \( 1 - 484. iT - 7.81e4T^{2} \)
7 \( 1 + 826.T + 8.23e5T^{2} \)
11 \( 1 + 4.55e3iT - 1.94e7T^{2} \)
13 \( 1 + 3.07e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.16e4T + 4.10e8T^{2} \)
19 \( 1 - 5.88e4iT - 8.93e8T^{2} \)
23 \( 1 + 1.86e4T + 3.40e9T^{2} \)
29 \( 1 + 2.10e5iT - 1.72e10T^{2} \)
31 \( 1 + 9.87e4T + 2.75e10T^{2} \)
37 \( 1 + 5.49e5iT - 9.49e10T^{2} \)
41 \( 1 - 5.59e5T + 1.94e11T^{2} \)
43 \( 1 + 5.03e5iT - 2.71e11T^{2} \)
47 \( 1 + 9.70e5T + 5.06e11T^{2} \)
53 \( 1 + 1.49e6iT - 1.17e12T^{2} \)
59 \( 1 - 4.15e5iT - 2.48e12T^{2} \)
61 \( 1 - 2.20e6iT - 3.14e12T^{2} \)
67 \( 1 + 1.95e6iT - 6.06e12T^{2} \)
71 \( 1 - 1.29e6T + 9.09e12T^{2} \)
73 \( 1 - 3.21e6T + 1.10e13T^{2} \)
79 \( 1 + 1.91e6T + 1.92e13T^{2} \)
83 \( 1 + 2.28e6iT - 2.71e13T^{2} \)
89 \( 1 + 3.17e6T + 4.42e13T^{2} \)
97 \( 1 - 2.37e6T + 8.07e13T^{2} \)
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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

−10.13842624071961833630044329726, −9.656353627457174254662087131134, −8.176962277892982983566652677540, −7.37207946247010632723145874189, −6.03210710339772995066579755157, −5.85467198882939796403203920812, −3.63334514874824119825912407016, −3.48497043348633997582827855642, −2.27912399210918182466820380204, −0.32553906803361694856833382737, 0.793234174394419984115129908722, 1.65375602737390772579458482924, 3.06067655934420743759286918160, 4.52859620238177416136875814775, 5.19758657384420384377154415091, 6.43848297610722601135184479233, 7.35436574002480343831064613935, 8.385565145521059342824424494242, 9.301446883260801984581245796956, 9.745921528182635441604174006426

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