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Properties

Label 2-384-16.13-c3-0-13
Degree $2$
Conductor $384$
Sign $0.937 + 0.349i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.12 − 2.12i)3-s + (−0.644 + 0.644i)5-s − 7.13i·7-s + 8.99i·9-s + (−25.4 + 25.4i)11-s + (14.6 + 14.6i)13-s + 2.73·15-s + 71.4·17-s + (43.6 + 43.6i)19-s + (−15.1 + 15.1i)21-s − 211. i·23-s + 124. i·25-s + (19.0 − 19.0i)27-s + (−5.84 − 5.84i)29-s − 107.·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.0576 + 0.0576i)5-s − 0.385i·7-s + 0.333i·9-s + (−0.697 + 0.697i)11-s + (0.311 + 0.311i)13-s + 0.0470·15-s + 1.01·17-s + (0.526 + 0.526i)19-s + (−0.157 + 0.157i)21-s − 1.92i·23-s + 0.993i·25-s + (0.136 − 0.136i)27-s + (−0.0374 − 0.0374i)29-s − 0.624·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.349i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.937 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.937 + 0.349i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.937 + 0.349i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.542771022\)
\(L(\frac12)\) \(\approx\) \(1.542771022\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.12 + 2.12i)T \)
good5 \( 1 + (0.644 - 0.644i)T - 125iT^{2} \)
7 \( 1 + 7.13iT - 343T^{2} \)
11 \( 1 + (25.4 - 25.4i)T - 1.33e3iT^{2} \)
13 \( 1 + (-14.6 - 14.6i)T + 2.19e3iT^{2} \)
17 \( 1 - 71.4T + 4.91e3T^{2} \)
19 \( 1 + (-43.6 - 43.6i)T + 6.85e3iT^{2} \)
23 \( 1 + 211. iT - 1.21e4T^{2} \)
29 \( 1 + (5.84 + 5.84i)T + 2.43e4iT^{2} \)
31 \( 1 + 107.T + 2.97e4T^{2} \)
37 \( 1 + (-184. + 184. i)T - 5.06e4iT^{2} \)
41 \( 1 - 360. iT - 6.89e4T^{2} \)
43 \( 1 + (-312. + 312. i)T - 7.95e4iT^{2} \)
47 \( 1 - 343.T + 1.03e5T^{2} \)
53 \( 1 + (-249. + 249. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-152. + 152. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-525. - 525. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-35.3 - 35.3i)T + 3.00e5iT^{2} \)
71 \( 1 + 784. iT - 3.57e5T^{2} \)
73 \( 1 + 800. iT - 3.89e5T^{2} \)
79 \( 1 - 548.T + 4.93e5T^{2} \)
83 \( 1 + (464. + 464. i)T + 5.71e5iT^{2} \)
89 \( 1 - 302. iT - 7.04e5T^{2} \)
97 \( 1 - 1.56e3T + 9.12e5T^{2} \)
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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

−10.77304673583952639357176723346, −10.17822100421223448860182677472, −9.034261816469724659933665713911, −7.80119383925446186277333192120, −7.22209018913718947409833003112, −6.05839883951786646476725934269, −5.07822635160053129188061643393, −3.85729887727194197042804304876, −2.33037664717628973575122912053, −0.830781720916230040608658713923, 0.866569757694081486461685394360, 2.78833726601556461970814913738, 3.91554534523459210500546295981, 5.40408899781036037480978305267, 5.77434413091635993790434737060, 7.27607750662776725580438482267, 8.197330079670865218700688364570, 9.241685033929372461572861687963, 10.08852269118536771224212544806, 11.01591352628006889942500929229

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