Properties

Degree $2$
Conductor $384$
Sign $-0.666 + 0.745i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.23 − 2i)3-s − 4·5-s + 8.94·7-s + (1.00 + 8.94i)9-s + 4.47·11-s − 17.8i·13-s + (8.94 + 8i)15-s − 17.8i·17-s + 20i·19-s + (−20.0 − 17.8i)21-s − 16i·23-s − 9·25-s + (15.6 − 22.0i)27-s − 52·29-s − 26.8·31-s + ⋯
L(s)  = 1  + (−0.745 − 0.666i)3-s − 0.800·5-s + 1.27·7-s + (0.111 + 0.993i)9-s + 0.406·11-s − 1.37i·13-s + (0.596 + 0.533i)15-s − 1.05i·17-s + 1.05i·19-s + (−0.952 − 0.851i)21-s − 0.695i·23-s − 0.359·25-s + (0.579 − 0.814i)27-s − 1.79·29-s − 0.865·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.666 + 0.745i$
Motivic weight: \(2\)
Character: $\chi_{384} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.666 + 0.745i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.357016 - 0.798312i\)
\(L(\frac12)\) \(\approx\) \(0.357016 - 0.798312i\)
\(L(2)\) 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 \( 1 + (2.23 + 2i)T \)
good5 \( 1 + 4T + 25T^{2} \)
7 \( 1 - 8.94T + 49T^{2} \)
11 \( 1 - 4.47T + 121T^{2} \)
13 \( 1 + 17.8iT - 169T^{2} \)
17 \( 1 + 17.8iT - 289T^{2} \)
19 \( 1 - 20iT - 361T^{2} \)
23 \( 1 + 16iT - 529T^{2} \)
29 \( 1 + 52T + 841T^{2} \)
31 \( 1 + 26.8T + 961T^{2} \)
37 \( 1 + 53.6iT - 1.36e3T^{2} \)
41 \( 1 - 35.7iT - 1.68e3T^{2} \)
43 \( 1 + 36iT - 1.84e3T^{2} \)
47 \( 1 + 64iT - 2.20e3T^{2} \)
53 \( 1 + 20T + 2.80e3T^{2} \)
59 \( 1 - 102.T + 3.48e3T^{2} \)
61 \( 1 + 17.8iT - 3.72e3T^{2} \)
67 \( 1 - 44iT - 4.48e3T^{2} \)
71 \( 1 + 80iT - 5.04e3T^{2} \)
73 \( 1 + 50T + 5.32e3T^{2} \)
79 \( 1 - 80.4T + 6.24e3T^{2} \)
83 \( 1 + 102.T + 6.88e3T^{2} \)
89 \( 1 + 160. iT - 7.92e3T^{2} \)
97 \( 1 - 50T + 9.40e3T^{2} \)
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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

−11.12723252326635143583321328382, −10.16020030078440039499839709051, −8.641865467853677943931564948805, −7.71835849348342913367122227067, −7.33118994576063656333441502810, −5.79440845256930210629715841074, −5.07387853715085123425158374915, −3.79397000014203190252817605218, −1.95017325878480092345137962446, −0.44329692545479391357174551610, 1.58883228402973684387131755115, 3.81165165516651943210809014544, 4.44477026422559753973472084666, 5.48568703416835118046212706929, 6.71020393402280904749582006875, 7.71089317366716823868673789761, 8.809227973886036052011330877948, 9.598364847075285612604971528630, 10.96550793636095942332280939734, 11.37157714469106289293062085740

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