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Properties

Label 2-384-48.35-c1-0-8
Degree $2$
Conductor $384$
Sign $0.998 + 0.0486i$
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  + (1.52 − 0.814i)3-s + (2.08 + 2.08i)5-s − 1.14·7-s + (1.67 − 2.48i)9-s + (1.67 − 1.67i)11-s + (−0.146 − 0.146i)13-s + (4.88 + 1.48i)15-s + 5.59i·17-s + (−1.48 + 1.48i)19-s + (−1.75 + 0.933i)21-s − 3.34i·23-s + 3.68i·25-s + (0.533 − 5.16i)27-s + (−3.51 + 3.51i)29-s − 5.83i·31-s + ⋯
L(s)  = 1  + (0.882 − 0.470i)3-s + (0.931 + 0.931i)5-s − 0.433·7-s + (0.558 − 0.829i)9-s + (0.504 − 0.504i)11-s + (−0.0405 − 0.0405i)13-s + (1.26 + 0.384i)15-s + 1.35i·17-s + (−0.341 + 0.341i)19-s + (−0.382 + 0.203i)21-s − 0.698i·23-s + 0.737i·25-s + (0.102 − 0.994i)27-s + (−0.652 + 0.652i)29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.998 + 0.0486i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.998 + 0.0486i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00865 - 0.0489117i\)
\(L(\frac12)\) \(\approx\) \(2.00865 - 0.0489117i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 0.814i)T \)
good5 \( 1 + (-2.08 - 2.08i)T + 5iT^{2} \)
7 \( 1 + 1.14T + 7T^{2} \)
11 \( 1 + (-1.67 + 1.67i)T - 11iT^{2} \)
13 \( 1 + (0.146 + 0.146i)T + 13iT^{2} \)
17 \( 1 - 5.59iT - 17T^{2} \)
19 \( 1 + (1.48 - 1.48i)T - 19iT^{2} \)
23 \( 1 + 3.34iT - 23T^{2} \)
29 \( 1 + (3.51 - 3.51i)T - 29iT^{2} \)
31 \( 1 + 5.83iT - 31T^{2} \)
37 \( 1 + (-4.83 + 4.83i)T - 37iT^{2} \)
41 \( 1 + 0.610T + 41T^{2} \)
43 \( 1 + (-1.48 - 1.48i)T + 43iT^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 + (-0.164 - 0.164i)T + 53iT^{2} \)
59 \( 1 + (9.05 - 9.05i)T - 59iT^{2} \)
61 \( 1 + (4.53 + 4.53i)T + 61iT^{2} \)
67 \( 1 + (-0.635 + 0.635i)T - 67iT^{2} \)
71 \( 1 - 6.90iT - 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 - 9.83iT - 79T^{2} \)
83 \( 1 + (8.09 + 8.09i)T + 83iT^{2} \)
89 \( 1 + 0.490T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−11.17688657475679132793741291059, −10.28108618760217909533252741465, −9.505263149989931049146510960775, −8.611449287230137118370234007666, −7.58516964201251407175194706823, −6.42883560351531077229758033043, −6.04982151947964235625684984650, −3.99006081777247294808988599521, −2.93724514502245192657080780505, −1.80122298426281180647347900878, 1.69902914385269716382626458823, 3.04602305193807420428062307323, 4.46689204175197002524389802077, 5.27808263481567227294316897803, 6.63719472092198756302100916060, 7.76350859467104471307339413909, 8.926620466076312570824926947220, 9.473901592580769185314901363141, 9.966099965868296919249950717585, 11.29766654609064562349428281583

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