Properties

Label 2-384-128.107-c2-0-41
Degree $2$
Conductor $384$
Sign $-0.293 + 0.955i$
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  + (−1.94 + 0.485i)2-s + (1.09 − 1.33i)3-s + (3.52 − 1.88i)4-s + (−1.67 − 5.53i)5-s + (−1.48 + 3.13i)6-s + (0.320 − 1.61i)7-s + (−5.93 + 5.36i)8-s + (−0.585 − 2.94i)9-s + (5.94 + 9.92i)10-s + (12.1 + 1.20i)11-s + (1.35 − 6.79i)12-s + (15.6 + 4.75i)13-s + (0.159 + 3.28i)14-s + (−9.25 − 3.83i)15-s + (8.91 − 13.2i)16-s + (−8.76 − 21.1i)17-s + ⋯
L(s)  = 1  + (−0.970 + 0.242i)2-s + (0.366 − 0.446i)3-s + (0.882 − 0.470i)4-s + (−0.335 − 1.10i)5-s + (−0.247 + 0.521i)6-s + (0.0458 − 0.230i)7-s + (−0.741 + 0.670i)8-s + (−0.0650 − 0.326i)9-s + (0.594 + 0.992i)10-s + (1.10 + 0.109i)11-s + (0.113 − 0.566i)12-s + (1.20 + 0.365i)13-s + (0.0114 + 0.234i)14-s + (−0.616 − 0.255i)15-s + (0.557 − 0.830i)16-s + (−0.515 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.293 + 0.955i$
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: \((2,\ 384,\ (\ :1),\ -0.293 + 0.955i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.670820 - 0.907609i\)
\(L(\frac12)\) \(\approx\) \(0.670820 - 0.907609i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.94 - 0.485i)T \)
3 \( 1 + (-1.09 + 1.33i)T \)
good5 \( 1 + (1.67 + 5.53i)T + (-20.7 + 13.8i)T^{2} \)
7 \( 1 + (-0.320 + 1.61i)T + (-45.2 - 18.7i)T^{2} \)
11 \( 1 + (-12.1 - 1.20i)T + (118. + 23.6i)T^{2} \)
13 \( 1 + (-15.6 - 4.75i)T + (140. + 93.8i)T^{2} \)
17 \( 1 + (8.76 + 21.1i)T + (-204. + 204. i)T^{2} \)
19 \( 1 + (-6.62 + 3.53i)T + (200. - 300. i)T^{2} \)
23 \( 1 + (7.23 + 4.83i)T + (202. + 488. i)T^{2} \)
29 \( 1 + (8.69 - 0.856i)T + (824. - 164. i)T^{2} \)
31 \( 1 + (14.9 - 14.9i)T - 961iT^{2} \)
37 \( 1 + (10.1 + 5.41i)T + (760. + 1.13e3i)T^{2} \)
41 \( 1 + (57.8 + 38.6i)T + (643. + 1.55e3i)T^{2} \)
43 \( 1 + (22.0 + 26.8i)T + (-360. + 1.81e3i)T^{2} \)
47 \( 1 + (0.701 + 1.69i)T + (-1.56e3 + 1.56e3i)T^{2} \)
53 \( 1 + (48.0 + 4.73i)T + (2.75e3 + 548. i)T^{2} \)
59 \( 1 + (24.8 + 81.9i)T + (-2.89e3 + 1.93e3i)T^{2} \)
61 \( 1 + (23.1 - 28.2i)T + (-725. - 3.64e3i)T^{2} \)
67 \( 1 + (3.61 + 2.96i)T + (875. + 4.40e3i)T^{2} \)
71 \( 1 + (-138. - 27.4i)T + (4.65e3 + 1.92e3i)T^{2} \)
73 \( 1 + (-31.0 + 6.18i)T + (4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (-20.5 + 49.6i)T + (-4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 + (27.0 + 50.6i)T + (-3.82e3 + 5.72e3i)T^{2} \)
89 \( 1 + (-93.9 - 140. i)T + (-3.03e3 + 7.31e3i)T^{2} \)
97 \( 1 + (28.7 + 28.7i)T + 9.40e3iT^{2} \)
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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

−10.87017850578788855068040605585, −9.475739934294581013979600987048, −8.932873218685350247154745915786, −8.306660073340279601285630660202, −7.18564400342081341931312578475, −6.44481496701192867634903065465, −5.06707359383759991824272106057, −3.63119529801680600962384489999, −1.78256156617256516508961894712, −0.69154483286829001782151206287, 1.67633991335317512578642515445, 3.21728537859878595074374278872, 3.86664744224197527259378754757, 6.03555509711908198647728016953, 6.77045325700465798678244371108, 7.950773126610272818902076265765, 8.659057264719671632050738321633, 9.575422158672197892195319746782, 10.53816191531371491673096945174, 11.11862887747109232307886117545

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