Properties

Label 2-384-128.107-c2-0-12
Degree $2$
Conductor $384$
Sign $0.998 - 0.0470i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.85 − 0.739i)2-s + (1.09 − 1.33i)3-s + (2.90 + 2.74i)4-s + (−1.54 − 5.10i)5-s + (−3.03 + 1.67i)6-s + (−2.29 + 11.5i)7-s + (−3.36 − 7.25i)8-s + (−0.585 − 2.94i)9-s + (−0.899 + 10.6i)10-s + (−8.91 − 0.877i)11-s + (6.87 − 0.868i)12-s + (19.0 + 5.76i)13-s + (12.8 − 19.7i)14-s + (−8.53 − 3.53i)15-s + (0.878 + 15.9i)16-s + (9.89 + 23.8i)17-s + ⋯
L(s)  = 1  + (−0.929 − 0.369i)2-s + (0.366 − 0.446i)3-s + (0.726 + 0.687i)4-s + (−0.309 − 1.02i)5-s + (−0.505 + 0.279i)6-s + (−0.328 + 1.65i)7-s + (−0.420 − 0.907i)8-s + (−0.0650 − 0.326i)9-s + (−0.0899 + 1.06i)10-s + (−0.810 − 0.0798i)11-s + (0.572 − 0.0723i)12-s + (1.46 + 0.443i)13-s + (0.915 − 1.41i)14-s + (−0.568 − 0.235i)15-s + (0.0549 + 0.998i)16-s + (0.582 + 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0470i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.998 - 0.0470i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.998 - 0.0470i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.08848 + 0.0256271i\)
\(L(\frac12)\) \(\approx\) \(1.08848 + 0.0256271i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.85 + 0.739i)T \)
3 \( 1 + (-1.09 + 1.33i)T \)
good5 \( 1 + (1.54 + 5.10i)T + (-20.7 + 13.8i)T^{2} \)
7 \( 1 + (2.29 - 11.5i)T + (-45.2 - 18.7i)T^{2} \)
11 \( 1 + (8.91 + 0.877i)T + (118. + 23.6i)T^{2} \)
13 \( 1 + (-19.0 - 5.76i)T + (140. + 93.8i)T^{2} \)
17 \( 1 + (-9.89 - 23.8i)T + (-204. + 204. i)T^{2} \)
19 \( 1 + (-9.15 + 4.89i)T + (200. - 300. i)T^{2} \)
23 \( 1 + (1.52 + 1.01i)T + (202. + 488. i)T^{2} \)
29 \( 1 + (3.82 - 0.376i)T + (824. - 164. i)T^{2} \)
31 \( 1 + (12.7 - 12.7i)T - 961iT^{2} \)
37 \( 1 + (-49.6 - 26.5i)T + (760. + 1.13e3i)T^{2} \)
41 \( 1 + (-38.8 - 25.9i)T + (643. + 1.55e3i)T^{2} \)
43 \( 1 + (40.3 + 49.1i)T + (-360. + 1.81e3i)T^{2} \)
47 \( 1 + (-6.54 - 15.7i)T + (-1.56e3 + 1.56e3i)T^{2} \)
53 \( 1 + (-54.0 - 5.32i)T + (2.75e3 + 548. i)T^{2} \)
59 \( 1 + (-20.3 - 66.9i)T + (-2.89e3 + 1.93e3i)T^{2} \)
61 \( 1 + (-70.7 + 86.2i)T + (-725. - 3.64e3i)T^{2} \)
67 \( 1 + (-100. - 82.5i)T + (875. + 4.40e3i)T^{2} \)
71 \( 1 + (-59.4 - 11.8i)T + (4.65e3 + 1.92e3i)T^{2} \)
73 \( 1 + (47.3 - 9.40i)T + (4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (4.12 - 9.95i)T + (-4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 + (74.6 + 139. i)T + (-3.82e3 + 5.72e3i)T^{2} \)
89 \( 1 + (-29.6 - 44.4i)T + (-3.03e3 + 7.31e3i)T^{2} \)
97 \( 1 + (17.1 + 17.1i)T + 9.40e3iT^{2} \)
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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

−11.26041565614717728901854220541, −10.01262138192290951826838656977, −8.937012251590736063892041290620, −8.562433966440965739542666421536, −7.922417816366341553178370736115, −6.42321204000746866309927105689, −5.53674119074915072511046927224, −3.72161795567552920713502936771, −2.47252967410044263714088696262, −1.19868132486209395943826067189, 0.74213453730427394094718116090, 2.85085198938887202051995262418, 3.82289888486541051819237269975, 5.48567899674084519683578648048, 6.76963899475354943661664177996, 7.49602740250954993676399050442, 8.095521356742923574830360009307, 9.513060072993628119962537584124, 10.14365094198349701062665257313, 10.92494414312238328431972787902

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