Properties

Label 2-384-32.5-c1-0-2
Degree $2$
Conductor $384$
Sign $0.942 - 0.334i$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 − 0.923i)3-s + (3.68 + 1.52i)5-s + (1.63 + 1.63i)7-s + (−0.707 + 0.707i)9-s + (−1.20 + 2.90i)11-s + (−3.30 + 1.36i)13-s − 3.99i·15-s − 0.511i·17-s + (0.254 − 0.105i)19-s + (0.885 − 2.13i)21-s + (3.17 − 3.17i)23-s + (7.73 + 7.73i)25-s + (0.923 + 0.382i)27-s + (−2.75 − 6.64i)29-s + 5.82·31-s + ⋯
L(s)  = 1  + (−0.220 − 0.533i)3-s + (1.64 + 0.683i)5-s + (0.618 + 0.618i)7-s + (−0.235 + 0.235i)9-s + (−0.363 + 0.876i)11-s + (−0.915 + 0.379i)13-s − 1.03i·15-s − 0.124i·17-s + (0.0582 − 0.0241i)19-s + (0.193 − 0.466i)21-s + (0.661 − 0.661i)23-s + (1.54 + 1.54i)25-s + (0.177 + 0.0736i)27-s + (−0.510 − 1.23i)29-s + 1.04·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.942 - 0.334i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.942 - 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62294 + 0.279084i\)
\(L(\frac12)\) \(\approx\) \(1.62294 + 0.279084i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.382 + 0.923i)T \)
good5 \( 1 + (-3.68 - 1.52i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.63 - 1.63i)T + 7iT^{2} \)
11 \( 1 + (1.20 - 2.90i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (3.30 - 1.36i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 0.511iT - 17T^{2} \)
19 \( 1 + (-0.254 + 0.105i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.17 + 3.17i)T - 23iT^{2} \)
29 \( 1 + (2.75 + 6.64i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 5.82T + 31T^{2} \)
37 \( 1 + (5.53 + 2.29i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.94 + 3.94i)T - 41iT^{2} \)
43 \( 1 + (1.30 - 3.15i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 9.41iT - 47T^{2} \)
53 \( 1 + (-2.50 + 6.04i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (13.3 + 5.52i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.35 - 8.10i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-1.11 - 2.69i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (1.26 + 1.26i)T + 71iT^{2} \)
73 \( 1 + (7.51 - 7.51i)T - 73iT^{2} \)
79 \( 1 + 0.709iT - 79T^{2} \)
83 \( 1 + (-1.40 + 0.583i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (0.0856 + 0.0856i)T + 89iT^{2} \)
97 \( 1 + 0.677T + 97T^{2} \)
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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

−11.42454076062817061890687266328, −10.32861606258480517274792530828, −9.736232587250721330776891590031, −8.721169232590178183713834660426, −7.40701051330783770289631817695, −6.63482188424249611854842034136, −5.63786365635020476668819630782, −4.83549056400048683595874271149, −2.55147044635118423005253543868, −1.95288875750756014829586317480, 1.32405508457457373895196580708, 2.94534053155541394262837593018, 4.70164201282435913042467655967, 5.32767507318919260114644334115, 6.23003248340620196822677825311, 7.61417635827055367918334671369, 8.784422317456594787823563122634, 9.515837605987435180161101964054, 10.39111393237680294383933963617, 10.96761858322445298127613024208

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