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Properties

Label 2-384-8.5-c9-0-17
Degree $2$
Conductor $384$
Sign $i$
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 + 2.35e3i·5-s − 1.31e3·7-s − 6.56e3·9-s + 1.26e4i·11-s + 1.49e5i·13-s − 1.91e5·15-s − 2.42e5·17-s + 7.13e5i·19-s − 1.06e5i·21-s − 6.34e5·23-s − 3.60e6·25-s − 5.31e5i·27-s − 9.19e5i·29-s + 5.06e6·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.68i·5-s − 0.206·7-s − 0.333·9-s + 0.260i·11-s + 1.45i·13-s − 0.974·15-s − 0.705·17-s + 1.25i·19-s − 0.119i·21-s − 0.472·23-s − 1.84·25-s − 0.192i·27-s − 0.241i·29-s + 0.984·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $i$
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: \((2,\ 384,\ (\ :9/2),\ i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.171056668\)
\(L(\frac12)\) \(\approx\) \(1.171056668\)
\(L(\frac{11}{2})\) 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 \( 1 - 2.35e3iT - 1.95e6T^{2} \)
7 \( 1 + 1.31e3T + 4.03e7T^{2} \)
11 \( 1 - 1.26e4iT - 2.35e9T^{2} \)
13 \( 1 - 1.49e5iT - 1.06e10T^{2} \)
17 \( 1 + 2.42e5T + 1.18e11T^{2} \)
19 \( 1 - 7.13e5iT - 3.22e11T^{2} \)
23 \( 1 + 6.34e5T + 1.80e12T^{2} \)
29 \( 1 + 9.19e5iT - 1.45e13T^{2} \)
31 \( 1 - 5.06e6T + 2.64e13T^{2} \)
37 \( 1 - 1.14e7iT - 1.29e14T^{2} \)
41 \( 1 + 8.44e6T + 3.27e14T^{2} \)
43 \( 1 - 1.27e7iT - 5.02e14T^{2} \)
47 \( 1 + 1.08e6T + 1.11e15T^{2} \)
53 \( 1 + 9.16e6iT - 3.29e15T^{2} \)
59 \( 1 + 2.45e6iT - 8.66e15T^{2} \)
61 \( 1 - 4.46e7iT - 1.16e16T^{2} \)
67 \( 1 - 2.55e8iT - 2.72e16T^{2} \)
71 \( 1 - 1.45e8T + 4.58e16T^{2} \)
73 \( 1 + 3.89e8T + 5.88e16T^{2} \)
79 \( 1 + 6.65e8T + 1.19e17T^{2} \)
83 \( 1 + 3.04e8iT - 1.86e17T^{2} \)
89 \( 1 - 3.38e8T + 3.50e17T^{2} \)
97 \( 1 - 1.28e9T + 7.60e17T^{2} \)
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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

−10.21730289428421039847007201054, −9.989073674554956424064705248135, −8.772724765623420906189966285407, −7.61379053400350189944009745330, −6.62204296078731781397751939194, −6.11898061551246277295459297211, −4.55040927806005006899726758321, −3.69780375276805099561938732101, −2.72931715810204188457901381318, −1.75820775679837980335629622032, 0.28316582757071225314109587507, 0.70355092951673281586299693642, 1.86588344513218760831140894470, 3.09441072449023122817742653857, 4.48076877927305423785801907932, 5.28207037786665857345116092853, 6.17279078098714351524985449835, 7.42104725128037991391009765887, 8.370003746269567343808198449870, 8.871006646009365575960319858175

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