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Properties

Label 2-384-32.29-c1-0-7
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: $\chi_{384} (337, \cdot )$
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

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

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