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Properties

Label 2-384-32.21-c1-0-1
Degree $2$
Conductor $384$
Sign $0.342 - 0.939i$
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 + (−1.48 + 3.58i)5-s + (−1.03 − 1.03i)7-s + (0.707 − 0.707i)9-s + (2.98 + 1.23i)11-s + (1.33 + 3.23i)13-s + 3.88i·15-s + 5.31i·17-s + (0.339 + 0.820i)19-s + (−1.35 − 0.561i)21-s + (−4.32 + 4.32i)23-s + (−7.13 − 7.13i)25-s + (0.382 − 0.923i)27-s + (5.78 − 2.39i)29-s + 1.42·31-s + ⋯
L(s)  = 1  + (0.533 − 0.220i)3-s + (−0.664 + 1.60i)5-s + (−0.392 − 0.392i)7-s + (0.235 − 0.235i)9-s + (0.901 + 0.373i)11-s + (0.371 + 0.897i)13-s + 1.00i·15-s + 1.28i·17-s + (0.0779 + 0.188i)19-s + (−0.296 − 0.122i)21-s + (−0.901 + 0.901i)23-s + (−1.42 − 1.42i)25-s + (0.0736 − 0.177i)27-s + (1.07 − 0.444i)29-s + 0.255·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.342 - 0.939i$
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: \((2,\ 384,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13038 + 0.791002i\)
\(L(\frac12)\) \(\approx\) \(1.13038 + 0.791002i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 - 3.58i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.03 + 1.03i)T + 7iT^{2} \)
11 \( 1 + (-2.98 - 1.23i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-1.33 - 3.23i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 5.31iT - 17T^{2} \)
19 \( 1 + (-0.339 - 0.820i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.32 - 4.32i)T - 23iT^{2} \)
29 \( 1 + (-5.78 + 2.39i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.42T + 31T^{2} \)
37 \( 1 + (-0.646 + 1.56i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.42 + 3.42i)T - 41iT^{2} \)
43 \( 1 + (6.50 + 2.69i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 + (-6.63 - 2.74i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.185 + 0.447i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.16 - 1.31i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-6.09 + 2.52i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (2.91 + 2.91i)T + 71iT^{2} \)
73 \( 1 + (-1.02 + 1.02i)T - 73iT^{2} \)
79 \( 1 + 12.8iT - 79T^{2} \)
83 \( 1 + (-6.41 - 15.4i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (0.991 + 0.991i)T + 89iT^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−11.59276922333786665921640535032, −10.50659409191759325218549217670, −9.880605806141277398271404602037, −8.650242643281726449341607016826, −7.64455479818729077182083299143, −6.80384350406467875105619428728, −6.24477131170599833740712936073, −3.99710562096989900015845894940, −3.56711757646203733431805144670, −2.03491706854413400195945800602, 0.935181752616556712503876199267, 2.97490556024877338177752701273, 4.21270083309247367775385421586, 5.06845603954461221898049666209, 6.31258245912176127169138675021, 7.76962022308878550446829583430, 8.532006115353510784181892795901, 9.118696681161435966590851173551, 9.981225259912079736178409063148, 11.37591279863501932458690650953

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