Properties

Label 2-384-128.107-c2-0-32
Degree $2$
Conductor $384$
Sign $0.306 + 0.951i$
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  + (−1.68 − 1.07i)2-s + (1.09 − 1.33i)3-s + (1.70 + 3.61i)4-s + (1.00 + 3.30i)5-s + (−3.29 + 1.08i)6-s + (0.212 − 1.06i)7-s + (1.00 − 7.93i)8-s + (−0.585 − 2.94i)9-s + (1.85 − 6.65i)10-s + (−9.91 − 0.976i)11-s + (6.71 + 1.69i)12-s + (3.75 + 1.13i)13-s + (−1.50 + 1.57i)14-s + (5.52 + 2.28i)15-s + (−10.2 + 12.3i)16-s + (−10.0 − 24.3i)17-s + ⋯
L(s)  = 1  + (−0.844 − 0.535i)2-s + (0.366 − 0.446i)3-s + (0.425 + 0.904i)4-s + (0.200 + 0.661i)5-s + (−0.548 + 0.180i)6-s + (0.0303 − 0.152i)7-s + (0.125 − 0.992i)8-s + (−0.0650 − 0.326i)9-s + (0.185 − 0.665i)10-s + (−0.901 − 0.0887i)11-s + (0.559 + 0.141i)12-s + (0.288 + 0.0876i)13-s + (−0.107 + 0.112i)14-s + (0.368 + 0.152i)15-s + (−0.637 + 0.770i)16-s + (−0.593 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.306 + 0.951i$
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: \((2,\ 384,\ (\ :1),\ 0.306 + 0.951i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.02759 - 0.748800i\)
\(L(\frac12)\) \(\approx\) \(1.02759 - 0.748800i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.68 + 1.07i)T \)
3 \( 1 + (-1.09 + 1.33i)T \)
good5 \( 1 + (-1.00 - 3.30i)T + (-20.7 + 13.8i)T^{2} \)
7 \( 1 + (-0.212 + 1.06i)T + (-45.2 - 18.7i)T^{2} \)
11 \( 1 + (9.91 + 0.976i)T + (118. + 23.6i)T^{2} \)
13 \( 1 + (-3.75 - 1.13i)T + (140. + 93.8i)T^{2} \)
17 \( 1 + (10.0 + 24.3i)T + (-204. + 204. i)T^{2} \)
19 \( 1 + (-24.5 + 13.1i)T + (200. - 300. i)T^{2} \)
23 \( 1 + (-35.3 - 23.6i)T + (202. + 488. i)T^{2} \)
29 \( 1 + (-30.6 + 3.01i)T + (824. - 164. i)T^{2} \)
31 \( 1 + (-23.0 + 23.0i)T - 961iT^{2} \)
37 \( 1 + (10.2 + 5.45i)T + (760. + 1.13e3i)T^{2} \)
41 \( 1 + (25.3 + 16.9i)T + (643. + 1.55e3i)T^{2} \)
43 \( 1 + (-47.3 - 57.6i)T + (-360. + 1.81e3i)T^{2} \)
47 \( 1 + (26.3 + 63.5i)T + (-1.56e3 + 1.56e3i)T^{2} \)
53 \( 1 + (84.9 + 8.36i)T + (2.75e3 + 548. i)T^{2} \)
59 \( 1 + (26.0 + 85.9i)T + (-2.89e3 + 1.93e3i)T^{2} \)
61 \( 1 + (-34.5 + 42.0i)T + (-725. - 3.64e3i)T^{2} \)
67 \( 1 + (-31.9 - 26.1i)T + (875. + 4.40e3i)T^{2} \)
71 \( 1 + (-58.3 - 11.6i)T + (4.65e3 + 1.92e3i)T^{2} \)
73 \( 1 + (4.82 - 0.959i)T + (4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (37.6 - 90.8i)T + (-4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 + (-36.6 - 68.5i)T + (-3.82e3 + 5.72e3i)T^{2} \)
89 \( 1 + (21.7 + 32.4i)T + (-3.03e3 + 7.31e3i)T^{2} \)
97 \( 1 + (117. + 117. i)T + 9.40e3iT^{2} \)
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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

−11.08936173178052744702118434884, −9.895006747262422650279155288264, −9.252857887061483615652593292713, −8.179797997015992138539913492846, −7.27063419979987050371999018515, −6.68628540814889979800989377000, −4.98253668002204462956992546562, −3.17020881572054655801156708588, −2.55699042849663771783481512013, −0.831491851740348734448425349159, 1.26097439393561151680853041758, 2.85698810697588103882306737456, 4.68815566700867191699415267752, 5.50616958647059380768294160645, 6.66784682503303207433354333217, 7.896848520523658061474769802012, 8.602749219521197684701672054007, 9.232765055893477592809859101485, 10.39885158787029685826585796873, 10.74492414308463980784451122854

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