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Properties

Label 2-384-48.5-c2-0-22
Degree $2$
Conductor $384$
Sign $0.325 + 0.945i$
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  + (2.06 − 2.17i)3-s + (3.17 − 3.17i)5-s + 6.03i·7-s + (−0.485 − 8.98i)9-s + (13.0 − 13.0i)11-s + (−6.39 + 6.39i)13-s + (−0.363 − 13.4i)15-s − 4.39i·17-s + (3.21 − 3.21i)19-s + (13.1 + 12.4i)21-s + 34.0·23-s + 4.78i·25-s + (−20.5 − 17.4i)27-s + (−27.9 − 27.9i)29-s − 7.90·31-s + ⋯
L(s)  = 1  + (0.687 − 0.725i)3-s + (0.635 − 0.635i)5-s + 0.862i·7-s + (−0.0539 − 0.998i)9-s + (1.18 − 1.18i)11-s + (−0.491 + 0.491i)13-s + (−0.0242 − 0.898i)15-s − 0.258i·17-s + (0.168 − 0.168i)19-s + (0.626 + 0.593i)21-s + 1.47·23-s + 0.191i·25-s + (−0.761 − 0.647i)27-s + (−0.964 − 0.964i)29-s − 0.255·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.325 + 0.945i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.325 + 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.97368 - 1.40721i\)
\(L(\frac12)\) \(\approx\) \(1.97368 - 1.40721i\)
\(L(2)\) 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 \( 1 + (-2.06 + 2.17i)T \)
good5 \( 1 + (-3.17 + 3.17i)T - 25iT^{2} \)
7 \( 1 - 6.03iT - 49T^{2} \)
11 \( 1 + (-13.0 + 13.0i)T - 121iT^{2} \)
13 \( 1 + (6.39 - 6.39i)T - 169iT^{2} \)
17 \( 1 + 4.39iT - 289T^{2} \)
19 \( 1 + (-3.21 + 3.21i)T - 361iT^{2} \)
23 \( 1 - 34.0T + 529T^{2} \)
29 \( 1 + (27.9 + 27.9i)T + 841iT^{2} \)
31 \( 1 + 7.90T + 961T^{2} \)
37 \( 1 + (20.0 + 20.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 45.1T + 1.68e3T^{2} \)
43 \( 1 + (-36.0 - 36.0i)T + 1.84e3iT^{2} \)
47 \( 1 - 5.08iT - 2.20e3T^{2} \)
53 \( 1 + (20.7 - 20.7i)T - 2.80e3iT^{2} \)
59 \( 1 + (-39.0 + 39.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (-49.8 + 49.8i)T - 3.72e3iT^{2} \)
67 \( 1 + (44.9 - 44.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 46.6T + 5.04e3T^{2} \)
73 \( 1 - 97.3iT - 5.32e3T^{2} \)
79 \( 1 + 40.1T + 6.24e3T^{2} \)
83 \( 1 + (-35.5 - 35.5i)T + 6.88e3iT^{2} \)
89 \( 1 - 69.6T + 7.92e3T^{2} \)
97 \( 1 - 61.0T + 9.40e3T^{2} \)
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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

−11.24788071415799549936319835710, −9.417102001978972958980736022030, −9.187448274805363272842646231162, −8.402287070708925238960340380198, −7.13841777652719492034600765027, −6.18828844874010495599371285162, −5.25725882883705498752306717157, −3.63399025627556965753369780022, −2.35912831202153136147952703107, −1.11158010955295498384716367260, 1.78961474901159798196075027766, 3.17963859484614198992591050746, 4.20872875176173470590837384771, 5.28841166084970564007713447466, 6.84221859503319360094503856253, 7.38631697192643855767894613963, 8.777689327157004508956744763543, 9.611169827103770746012997509582, 10.28639031740528248312997460467, 10.91855486272218502820292272005

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