Properties

Degree $2$
Conductor $384$
Sign $0.296 + 0.955i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.86 − 0.888i)3-s − 8.59i·5-s + 10.9·7-s + (7.41 − 5.09i)9-s + 2.75i·11-s − 4.43·13-s + (−7.63 − 24.6i)15-s + 25.4i·17-s − 17.5·19-s + (31.3 − 9.71i)21-s − 17.5i·23-s − 48.8·25-s + (16.7 − 21.1i)27-s − 19.6i·29-s + 2.58·31-s + ⋯
L(s)  = 1  + (0.955 − 0.296i)3-s − 1.71i·5-s + 1.56·7-s + (0.824 − 0.565i)9-s + 0.250i·11-s − 0.341·13-s + (−0.509 − 1.64i)15-s + 1.49i·17-s − 0.923·19-s + (1.49 − 0.462i)21-s − 0.762i·23-s − 1.95·25-s + (0.619 − 0.784i)27-s − 0.676i·29-s + 0.0833·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.296 + 0.955i$
Motivic weight: \(2\)
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.296 + 0.955i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.15169 - 1.58538i\)
\(L(\frac12)\) \(\approx\) \(2.15169 - 1.58538i\)
\(L(2)\) 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 \( 1 + (-2.86 + 0.888i)T \)
good5 \( 1 + 8.59iT - 25T^{2} \)
7 \( 1 - 10.9T + 49T^{2} \)
11 \( 1 - 2.75iT - 121T^{2} \)
13 \( 1 + 4.43T + 169T^{2} \)
17 \( 1 - 25.4iT - 289T^{2} \)
19 \( 1 + 17.5T + 361T^{2} \)
23 \( 1 + 17.5iT - 529T^{2} \)
29 \( 1 + 19.6iT - 841T^{2} \)
31 \( 1 - 2.58T + 961T^{2} \)
37 \( 1 - 7.73T + 1.36e3T^{2} \)
41 \( 1 - 58.0iT - 1.68e3T^{2} \)
43 \( 1 - 42.1T + 1.84e3T^{2} \)
47 \( 1 - 17.4iT - 2.20e3T^{2} \)
53 \( 1 - 69.0iT - 2.80e3T^{2} \)
59 \( 1 + 50.5iT - 3.48e3T^{2} \)
61 \( 1 + 32.5T + 3.72e3T^{2} \)
67 \( 1 - 48.0T + 4.48e3T^{2} \)
71 \( 1 + 22.1iT - 5.04e3T^{2} \)
73 \( 1 + 27.0T + 5.32e3T^{2} \)
79 \( 1 + 97.4T + 6.24e3T^{2} \)
83 \( 1 - 59.5iT - 6.88e3T^{2} \)
89 \( 1 + 110. iT - 7.92e3T^{2} \)
97 \( 1 - 55.1T + 9.40e3T^{2} \)
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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

−10.96164032584876648122143754174, −9.785080941412448886194895085865, −8.740378100530993742282106795942, −8.300693045161737115583847125912, −7.67670527712465876620856738100, −6.05069925480381234274437082962, −4.65069275089407008191282685703, −4.24713590022207372357145338015, −2.11886106713709965700863847320, −1.21971313296375126571640131338, 2.03150728254347334979019145380, 2.94714347623852754236268836460, 4.15790049549712108358714070917, 5.35919858772771544451231580974, 6.98469385856200234031243788824, 7.50437969502016079636052695909, 8.449265472493335715940035344233, 9.512621761713400865167795879734, 10.54884927630158757974316847707, 11.06244458583045468797194859986

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