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Properties

Label 2-384-48.29-c2-0-16
Degree $2$
Conductor $384$
Sign $0.978 - 0.204i$
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  + (2.77 − 1.13i)3-s + (6.28 + 6.28i)5-s − 1.64i·7-s + (6.43 − 6.29i)9-s + (−4.75 − 4.75i)11-s + (9.35 + 9.35i)13-s + (24.5 + 10.3i)15-s − 11.4i·17-s + (8.58 + 8.58i)19-s + (−1.86 − 4.57i)21-s − 16.2·23-s + 54.0i·25-s + (10.7 − 24.7i)27-s + (−10.7 + 10.7i)29-s + 6.35·31-s + ⋯
L(s)  = 1  + (0.926 − 0.377i)3-s + (1.25 + 1.25i)5-s − 0.235i·7-s + (0.715 − 0.699i)9-s + (−0.432 − 0.432i)11-s + (0.719 + 0.719i)13-s + (1.63 + 0.689i)15-s − 0.675i·17-s + (0.451 + 0.451i)19-s + (−0.0887 − 0.217i)21-s − 0.706·23-s + 2.16i·25-s + (0.398 − 0.917i)27-s + (−0.370 + 0.370i)29-s + 0.204·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.978 - 0.204i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.978 - 0.204i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.85975 + 0.296113i\)
\(L(\frac12)\) \(\approx\) \(2.85975 + 0.296113i\)
\(L(2)\) 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 \( 1 + (-2.77 + 1.13i)T \)
good5 \( 1 + (-6.28 - 6.28i)T + 25iT^{2} \)
7 \( 1 + 1.64iT - 49T^{2} \)
11 \( 1 + (4.75 + 4.75i)T + 121iT^{2} \)
13 \( 1 + (-9.35 - 9.35i)T + 169iT^{2} \)
17 \( 1 + 11.4iT - 289T^{2} \)
19 \( 1 + (-8.58 - 8.58i)T + 361iT^{2} \)
23 \( 1 + 16.2T + 529T^{2} \)
29 \( 1 + (10.7 - 10.7i)T - 841iT^{2} \)
31 \( 1 - 6.35T + 961T^{2} \)
37 \( 1 + (27.2 - 27.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 1.98T + 1.68e3T^{2} \)
43 \( 1 + (-19.4 + 19.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 74.9iT - 2.20e3T^{2} \)
53 \( 1 + (-4.00 - 4.00i)T + 2.80e3iT^{2} \)
59 \( 1 + (27.9 + 27.9i)T + 3.48e3iT^{2} \)
61 \( 1 + (39.2 + 39.2i)T + 3.72e3iT^{2} \)
67 \( 1 + (-68.6 - 68.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 40.6T + 5.04e3T^{2} \)
73 \( 1 + 59.0iT - 5.32e3T^{2} \)
79 \( 1 - 17.3T + 6.24e3T^{2} \)
83 \( 1 + (75.1 - 75.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 78.8T + 7.92e3T^{2} \)
97 \( 1 + 38.8T + 9.40e3T^{2} \)
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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

−10.94945174473273421568914990110, −10.12101207739662515981234783270, −9.425224779561931137022323746179, −8.413734458390476311832356385266, −7.24133111021795971409645261876, −6.58704575376949049822873977864, −5.57527494167273533557374821294, −3.73271571081024151529926470469, −2.74477896380149761912823471855, −1.67782246281251738608200624864, 1.44203153880854921599146985678, 2.59114828121415343592799496696, 4.14262176404159059512732154272, 5.21463551383421862344424100401, 6.03086987071935644504808371691, 7.67652012773621536142831878974, 8.538722613074217538958173386213, 9.227688615071741267781314522852, 9.954170899256512322888140911418, 10.75253420713670618799223564849

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