Properties

Degree $2$
Conductor $384$
Sign $-0.208 + 0.977i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s + (−1.69 − 1.69i)5-s + 5.74·7-s + 2.99i·9-s + (−5.59 + 5.59i)11-s + (13.5 − 13.5i)13-s + 4.16i·15-s + 19.7·17-s + (−21.6 − 21.6i)19-s + (−7.03 − 7.03i)21-s − 24.9·23-s − 19.2i·25-s + (3.67 − 3.67i)27-s + (−1.50 + 1.50i)29-s − 2.20i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.339 − 0.339i)5-s + 0.820·7-s + 0.333i·9-s + (−0.508 + 0.508i)11-s + (1.04 − 1.04i)13-s + 0.277i·15-s + 1.15·17-s + (−1.14 − 1.14i)19-s + (−0.334 − 0.334i)21-s − 1.08·23-s − 0.768i·25-s + (0.136 − 0.136i)27-s + (−0.0519 + 0.0519i)29-s − 0.0709i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.208 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.208 + 0.977i$
Motivic weight: \(2\)
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.208 + 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.779532 - 0.963399i\)
\(L(\frac12)\) \(\approx\) \(0.779532 - 0.963399i\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
good5 \( 1 + (1.69 + 1.69i)T + 25iT^{2} \)
7 \( 1 - 5.74T + 49T^{2} \)
11 \( 1 + (5.59 - 5.59i)T - 121iT^{2} \)
13 \( 1 + (-13.5 + 13.5i)T - 169iT^{2} \)
17 \( 1 - 19.7T + 289T^{2} \)
19 \( 1 + (21.6 + 21.6i)T + 361iT^{2} \)
23 \( 1 + 24.9T + 529T^{2} \)
29 \( 1 + (1.50 - 1.50i)T - 841iT^{2} \)
31 \( 1 + 2.20iT - 961T^{2} \)
37 \( 1 + (27.6 + 27.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 51.3iT - 1.68e3T^{2} \)
43 \( 1 + (-21.4 + 21.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 76.5iT - 2.20e3T^{2} \)
53 \( 1 + (-56.5 - 56.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (48.0 - 48.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (-51.5 + 51.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (-63.4 - 63.4i)T + 4.48e3iT^{2} \)
71 \( 1 + 43.4T + 5.04e3T^{2} \)
73 \( 1 + 73.9iT - 5.32e3T^{2} \)
79 \( 1 + 4.12iT - 6.24e3T^{2} \)
83 \( 1 + (-38.4 - 38.4i)T + 6.88e3iT^{2} \)
89 \( 1 - 52.9iT - 7.92e3T^{2} \)
97 \( 1 - 23.1T + 9.40e3T^{2} \)
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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

−10.79178906178495385003234914251, −10.30175163871512698354057721930, −8.717593746645134277508562683083, −8.083301720211175995467118236876, −7.22428600766838224433308799257, −5.91239589199145708314170464792, −5.08132059151959270186729179710, −3.90055445157024729805950608697, −2.17056060977863330979155897332, −0.60722037159671060640238112686, 1.57090512421634719161356101859, 3.44291341496191054472297469817, 4.38438942670500530302075760885, 5.62750761735317760219159049749, 6.45906885105951621303759956757, 7.85838703300300745503554189194, 8.439153806388862224125888203629, 9.718137025581353209829645562419, 10.64263494718588645760083662068, 11.31250790606759406493297007280

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