Properties

Label 2-384-24.5-c2-0-10
Degree $2$
Conductor $384$
Sign $0.371 - 0.928i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.75 + 1.18i)3-s − 3.68·5-s + 5.43·7-s + (6.21 − 6.51i)9-s − 1.16·11-s − 14.4i·13-s + (10.1 − 4.35i)15-s − 1.71i·17-s + 28.8i·19-s + (−14.9 + 6.42i)21-s + 35.4i·23-s − 11.4·25-s + (−9.43 + 25.2i)27-s + 33.6·29-s + 55.5·31-s + ⋯
L(s)  = 1  + (−0.919 + 0.393i)3-s − 0.736·5-s + 0.776·7-s + (0.690 − 0.723i)9-s − 0.105·11-s − 1.10i·13-s + (0.677 − 0.290i)15-s − 0.100i·17-s + 1.51i·19-s + (−0.714 + 0.305i)21-s + 1.54i·23-s − 0.456·25-s + (−0.349 + 0.936i)27-s + 1.16·29-s + 1.79·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.371 - 0.928i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.371 - 0.928i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.837865 + 0.567065i\)
\(L(\frac12)\) \(\approx\) \(0.837865 + 0.567065i\)
\(L(2)\) 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 \( 1 + (2.75 - 1.18i)T \)
good5 \( 1 + 3.68T + 25T^{2} \)
7 \( 1 - 5.43T + 49T^{2} \)
11 \( 1 + 1.16T + 121T^{2} \)
13 \( 1 + 14.4iT - 169T^{2} \)
17 \( 1 + 1.71iT - 289T^{2} \)
19 \( 1 - 28.8iT - 361T^{2} \)
23 \( 1 - 35.4iT - 529T^{2} \)
29 \( 1 - 33.6T + 841T^{2} \)
31 \( 1 - 55.5T + 961T^{2} \)
37 \( 1 - 30.4iT - 1.36e3T^{2} \)
41 \( 1 - 63.4iT - 1.68e3T^{2} \)
43 \( 1 - 43.0iT - 1.84e3T^{2} \)
47 \( 1 + 61.5iT - 2.20e3T^{2} \)
53 \( 1 - 56.3T + 2.80e3T^{2} \)
59 \( 1 + 49.6T + 3.48e3T^{2} \)
61 \( 1 + 4.73iT - 3.72e3T^{2} \)
67 \( 1 + 7.08iT - 4.48e3T^{2} \)
71 \( 1 - 96.9iT - 5.04e3T^{2} \)
73 \( 1 - 68.5T + 5.32e3T^{2} \)
79 \( 1 + 9.72T + 6.24e3T^{2} \)
83 \( 1 - 38.9T + 6.88e3T^{2} \)
89 \( 1 + 0.675iT - 7.92e3T^{2} \)
97 \( 1 - 86.1T + 9.40e3T^{2} \)
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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

−11.60175407207486593375654422141, −10.35559078224675977351957462420, −9.852065265440152081560368535278, −8.226369814384602222724041426017, −7.75852197164604372508225293995, −6.37650039321578909323561607975, −5.38668944911650480458504630082, −4.48201607378176314562107557056, −3.34605824989184909877801957159, −1.15413403952308972479848764188, 0.61829304095122317236775229696, 2.28850714727856581605985229117, 4.31646764183003457234512881129, 4.86394310156282979361975467274, 6.30774330932002713467616251149, 7.08055565965471839157549162329, 8.042400511287563774393105979989, 8.972551797919170846014834098319, 10.39825699014697183561464384227, 11.08717348940724422728110493643

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