Properties

Label 2-384-3.2-c4-0-2
Degree $2$
Conductor $384$
Sign $-0.929 + 0.369i$
Analytic cond. $39.6940$
Root an. cond. $6.30032$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.32 + 8.36i)3-s − 25.4i·5-s − 36.1·7-s + (−58.9 + 55.5i)9-s + 188. i·11-s + 223.·13-s + (212. − 84.5i)15-s − 271. i·17-s − 255.·19-s + (−119. − 302. i)21-s + 308. i·23-s − 23.4·25-s + (−660. − 308. i)27-s − 1.04e3i·29-s − 1.13e3·31-s + ⋯
L(s)  = 1  + (0.369 + 0.929i)3-s − 1.01i·5-s − 0.737·7-s + (−0.727 + 0.685i)9-s + 1.55i·11-s + 1.32·13-s + (0.946 − 0.375i)15-s − 0.938i·17-s − 0.708·19-s + (−0.272 − 0.685i)21-s + 0.582i·23-s − 0.0374·25-s + (−0.906 − 0.423i)27-s − 1.24i·29-s − 1.17·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.929 + 0.369i$
Analytic conductor: \(39.6940\)
Root analytic conductor: \(6.30032\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :2),\ -0.929 + 0.369i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.2281933434\)
\(L(\frac12)\) \(\approx\) \(0.2281933434\)
\(L(3)\) 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 \( 1 + (-3.32 - 8.36i)T \)
good5 \( 1 + 25.4iT - 625T^{2} \)
7 \( 1 + 36.1T + 2.40e3T^{2} \)
11 \( 1 - 188. iT - 1.46e4T^{2} \)
13 \( 1 - 223.T + 2.85e4T^{2} \)
17 \( 1 + 271. iT - 8.35e4T^{2} \)
19 \( 1 + 255.T + 1.30e5T^{2} \)
23 \( 1 - 308. iT - 2.79e5T^{2} \)
29 \( 1 + 1.04e3iT - 7.07e5T^{2} \)
31 \( 1 + 1.13e3T + 9.23e5T^{2} \)
37 \( 1 + 1.42e3T + 1.87e6T^{2} \)
41 \( 1 - 1.58e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.58e3T + 3.41e6T^{2} \)
47 \( 1 - 430. iT - 4.87e6T^{2} \)
53 \( 1 - 316. iT - 7.89e6T^{2} \)
59 \( 1 + 5.79e3iT - 1.21e7T^{2} \)
61 \( 1 + 7.29e3T + 1.38e7T^{2} \)
67 \( 1 + 4.83e3T + 2.01e7T^{2} \)
71 \( 1 - 733. iT - 2.54e7T^{2} \)
73 \( 1 + 7.14e3T + 2.83e7T^{2} \)
79 \( 1 - 7.04e3T + 3.89e7T^{2} \)
83 \( 1 - 4.73e3iT - 4.74e7T^{2} \)
89 \( 1 + 581. iT - 6.27e7T^{2} \)
97 \( 1 - 7.16e3T + 8.85e7T^{2} \)
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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

−11.12098460441275938834905712257, −10.04501193666394790183632277214, −9.403578138402520327347150217263, −8.744486708714644874791617909096, −7.69397497740313236029358399655, −6.37650855345660177589174358000, −5.13436082067189379615151224821, −4.36899525269998326713294561204, −3.30942721297923672935006885903, −1.77488623867967708863692686579, 0.05849399827237533179645134362, 1.58322568156190803182929622275, 3.11762477410701500516408052284, 3.55381800913929082970511290083, 5.88442303962995413485987542529, 6.36547613095876192158722987640, 7.20438094631260374462899950676, 8.532339438351251336506765442029, 8.845409700352860521101360969726, 10.61819819377219771458739050432

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