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Properties

Label 2-384-48.35-c3-0-6
Degree $2$
Conductor $384$
Sign $-0.517 - 0.855i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.16 + 0.563i)3-s + (−13.1 − 13.1i)5-s − 13.2·7-s + (26.3 + 5.82i)9-s + (−24.0 + 24.0i)11-s + (30.7 + 30.7i)13-s + (−60.5 − 75.4i)15-s − 56.8i·17-s + (−74.2 + 74.2i)19-s + (−68.5 − 7.47i)21-s + 21.7i·23-s + 221. i·25-s + (132. + 44.9i)27-s + (−102. + 102. i)29-s + 219. i·31-s + ⋯
L(s)  = 1  + (0.994 + 0.108i)3-s + (−1.17 − 1.17i)5-s − 0.716·7-s + (0.976 + 0.215i)9-s + (−0.660 + 0.660i)11-s + (0.656 + 0.656i)13-s + (−1.04 − 1.29i)15-s − 0.811i·17-s + (−0.896 + 0.896i)19-s + (−0.712 − 0.0777i)21-s + 0.197i·23-s + 1.77i·25-s + (0.947 + 0.320i)27-s + (−0.657 + 0.657i)29-s + 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.517 - 0.855i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.517 - 0.855i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7129653917\)
\(L(\frac12)\) \(\approx\) \(0.7129653917\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.16 - 0.563i)T \)
good5 \( 1 + (13.1 + 13.1i)T + 125iT^{2} \)
7 \( 1 + 13.2T + 343T^{2} \)
11 \( 1 + (24.0 - 24.0i)T - 1.33e3iT^{2} \)
13 \( 1 + (-30.7 - 30.7i)T + 2.19e3iT^{2} \)
17 \( 1 + 56.8iT - 4.91e3T^{2} \)
19 \( 1 + (74.2 - 74.2i)T - 6.85e3iT^{2} \)
23 \( 1 - 21.7iT - 1.21e4T^{2} \)
29 \( 1 + (102. - 102. i)T - 2.43e4iT^{2} \)
31 \( 1 - 219. iT - 2.97e4T^{2} \)
37 \( 1 + (83.3 - 83.3i)T - 5.06e4iT^{2} \)
41 \( 1 + 7.92T + 6.89e4T^{2} \)
43 \( 1 + (-153. - 153. i)T + 7.95e4iT^{2} \)
47 \( 1 + 208.T + 1.03e5T^{2} \)
53 \( 1 + (390. + 390. i)T + 1.48e5iT^{2} \)
59 \( 1 + (221. - 221. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-416. - 416. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-284. + 284. i)T - 3.00e5iT^{2} \)
71 \( 1 + 26.9iT - 3.57e5T^{2} \)
73 \( 1 + 839. iT - 3.89e5T^{2} \)
79 \( 1 - 556. iT - 4.93e5T^{2} \)
83 \( 1 + (400. + 400. i)T + 5.71e5iT^{2} \)
89 \( 1 + 974.T + 7.04e5T^{2} \)
97 \( 1 + 1.55e3T + 9.12e5T^{2} \)
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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

−11.27835872844644737564878836826, −10.08151709064829950650259507971, −9.187871398289878969685774538235, −8.489319536339174192167154789034, −7.72708866826936044710458321440, −6.76772057467236297651455128612, −5.03415385528870835500677871960, −4.15270620112957488034455656157, −3.23564639034158830227754239588, −1.56895306947396642675473413607, 0.20937071375927643282688423530, 2.51187740055850651915511359915, 3.36980816558809304313250792619, 4.10973911582852236183959357017, 6.07502438503125686225405363332, 6.97876185451965821814990709453, 7.926754761823601517681899550270, 8.465008481315646647064400299457, 9.723881257797269851214293971756, 10.73225188744395875554639432081

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