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Properties

Label 2-384-128.101-c1-0-12
Degree $2$
Conductor $384$
Sign $-0.225 + 0.974i$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.788 − 1.17i)2-s + (−0.881 + 0.471i)3-s + (−0.756 + 1.85i)4-s + (−0.453 − 0.552i)5-s + (1.24 + 0.663i)6-s + (−1.44 + 0.965i)7-s + (2.76 − 0.572i)8-s + (0.555 − 0.831i)9-s + (−0.290 + 0.967i)10-s + (2.30 − 0.698i)11-s + (−0.206 − 1.98i)12-s + (−1.74 − 1.42i)13-s + (2.27 + 0.935i)14-s + (0.660 + 0.273i)15-s + (−2.85 − 2.79i)16-s + (5.03 − 2.08i)17-s + ⋯
L(s)  = 1  + (−0.557 − 0.830i)2-s + (−0.509 + 0.272i)3-s + (−0.378 + 0.925i)4-s + (−0.202 − 0.247i)5-s + (0.509 + 0.270i)6-s + (−0.546 + 0.365i)7-s + (0.979 − 0.202i)8-s + (0.185 − 0.277i)9-s + (−0.0920 + 0.306i)10-s + (0.693 − 0.210i)11-s + (−0.0594 − 0.574i)12-s + (−0.482 − 0.396i)13-s + (0.607 + 0.249i)14-s + (0.170 + 0.0706i)15-s + (−0.714 − 0.699i)16-s + (1.22 − 0.505i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.225 + 0.974i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.225 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.416848 - 0.524535i\)
\(L(\frac12)\) \(\approx\) \(0.416848 - 0.524535i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.788 + 1.17i)T \)
3 \( 1 + (0.881 - 0.471i)T \)
good5 \( 1 + (0.453 + 0.552i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (1.44 - 0.965i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-2.30 + 0.698i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (1.74 + 1.42i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (-5.03 + 2.08i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (0.389 + 3.95i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (1.78 - 0.354i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-2.69 + 8.88i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (-0.759 - 0.759i)T + 31iT^{2} \)
37 \( 1 + (-0.701 - 0.0690i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (1.73 + 8.73i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (3.05 + 1.63i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (-3.60 - 8.70i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (0.661 + 2.18i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (-7.60 + 6.23i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (-0.233 - 0.437i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (1.98 + 3.72i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (-2.72 - 4.07i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-0.964 - 0.644i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (0.799 - 1.93i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (4.07 - 0.401i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (-10.4 - 2.08i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (-13.4 - 13.4i)T + 97iT^{2} \)
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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

−11.10772991441405407145388926353, −10.05227348488055358900866411751, −9.532766692087827986749244611816, −8.542892438076758950356472002620, −7.51820669957814479712069424612, −6.30534498434057780870788387804, −4.99210498966444991800829744005, −3.86105915271229382391084884345, −2.63931399645700086914689133397, −0.64074510316808785523432583663, 1.40364833818679942832555625083, 3.69017687202608257959065063182, 5.05030145982964844280257642105, 6.13348760940492421008744556763, 6.91600040640699316522396833389, 7.65442669018545944810014701832, 8.742633996928597782827052131460, 9.889837337122241865191430322112, 10.35210905926317265386473094526, 11.54522217537844559360696483691

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