Properties

Label 2-384-32.21-c1-0-5
Degree $2$
Conductor $384$
Sign $-0.676 + 0.736i$
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  + (−0.923 + 0.382i)3-s + (−1.35 + 3.27i)5-s + (−2.48 − 2.48i)7-s + (0.707 − 0.707i)9-s + (−0.420 − 0.174i)11-s + (−1.98 − 4.80i)13-s − 3.54i·15-s − 4.75i·17-s + (0.402 + 0.971i)19-s + (3.24 + 1.34i)21-s + (−0.739 + 0.739i)23-s + (−5.36 − 5.36i)25-s + (−0.382 + 0.923i)27-s + (−0.153 + 0.0634i)29-s − 8.57·31-s + ⋯
L(s)  = 1  + (−0.533 + 0.220i)3-s + (−0.607 + 1.46i)5-s + (−0.939 − 0.939i)7-s + (0.235 − 0.235i)9-s + (−0.126 − 0.0525i)11-s + (−0.551 − 1.33i)13-s − 0.916i·15-s − 1.15i·17-s + (0.0923 + 0.222i)19-s + (0.708 + 0.293i)21-s + (−0.154 + 0.154i)23-s + (−1.07 − 1.07i)25-s + (−0.0736 + 0.177i)27-s + (−0.0284 + 0.0117i)29-s − 1.54·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.676 + 0.736i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.676 + 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0717155 - 0.163297i\)
\(L(\frac12)\) \(\approx\) \(0.0717155 - 0.163297i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.923 - 0.382i)T \)
good5 \( 1 + (1.35 - 3.27i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (2.48 + 2.48i)T + 7iT^{2} \)
11 \( 1 + (0.420 + 0.174i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.98 + 4.80i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 4.75iT - 17T^{2} \)
19 \( 1 + (-0.402 - 0.971i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.739 - 0.739i)T - 23iT^{2} \)
29 \( 1 + (0.153 - 0.0634i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 8.57T + 31T^{2} \)
37 \( 1 + (2.67 - 6.46i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.39 - 1.39i)T - 41iT^{2} \)
43 \( 1 + (2.84 + 1.18i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 0.715iT - 47T^{2} \)
53 \( 1 + (10.4 + 4.32i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (2.09 - 5.05i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (2.81 - 1.16i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-5.39 + 2.23i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-8.26 - 8.26i)T + 71iT^{2} \)
73 \( 1 + (-4.37 + 4.37i)T - 73iT^{2} \)
79 \( 1 + 9.46iT - 79T^{2} \)
83 \( 1 + (2.85 + 6.88i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-8.60 - 8.60i)T + 89iT^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−10.85351343175043742321306674309, −10.27959610348878731157377398979, −9.585603736770996303475214471322, −7.81857590373202275538129282104, −7.16731884180114642059970591076, −6.42626897754163820114693878314, −5.15144762137613635890904136323, −3.65553265260804112979988199090, −2.98693695752313264517577875655, −0.12104513945157395420253498719, 1.88459933174850611447266379964, 3.82470001193962696688993251644, 4.90190620814752890068135691901, 5.82864257940499872831096525721, 6.87319710081750878581259295107, 8.081489689517339693010305934969, 9.039075030916540959654868796069, 9.543006459725014519099860939480, 10.97600267456089109912877707948, 11.92664393882560677484916882295

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