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Properties

Label 2-384-128.37-c1-0-21
Degree $2$
Conductor $384$
Sign $0.597 + 0.801i$
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  + (−1.14 − 0.834i)2-s + (0.471 + 0.881i)3-s + (0.608 + 1.90i)4-s + (3.04 − 2.50i)5-s + (0.197 − 1.40i)6-s + (1.75 − 1.17i)7-s + (0.894 − 2.68i)8-s + (−0.555 + 0.831i)9-s + (−5.56 + 0.314i)10-s + (−0.491 − 1.62i)11-s + (−1.39 + 1.43i)12-s + (−0.848 + 1.03i)13-s + (−2.98 − 0.124i)14-s + (3.64 + 1.50i)15-s + (−3.25 + 2.31i)16-s + (2.25 − 0.935i)17-s + ⋯
L(s)  = 1  + (−0.807 − 0.589i)2-s + (0.272 + 0.509i)3-s + (0.304 + 0.952i)4-s + (1.36 − 1.11i)5-s + (0.0805 − 0.571i)6-s + (0.663 − 0.443i)7-s + (0.316 − 0.948i)8-s + (−0.185 + 0.277i)9-s + (−1.76 + 0.0994i)10-s + (−0.148 − 0.488i)11-s + (−0.402 + 0.414i)12-s + (−0.235 + 0.286i)13-s + (−0.796 − 0.0332i)14-s + (0.940 + 0.389i)15-s + (−0.814 + 0.579i)16-s + (0.547 − 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.597 + 0.801i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.597 + 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16874 - 0.586743i\)
\(L(\frac12)\) \(\approx\) \(1.16874 - 0.586743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 + 0.834i)T \)
3 \( 1 + (-0.471 - 0.881i)T \)
good5 \( 1 + (-3.04 + 2.50i)T + (0.975 - 4.90i)T^{2} \)
7 \( 1 + (-1.75 + 1.17i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (0.491 + 1.62i)T + (-9.14 + 6.11i)T^{2} \)
13 \( 1 + (0.848 - 1.03i)T + (-2.53 - 12.7i)T^{2} \)
17 \( 1 + (-2.25 + 0.935i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (4.51 - 0.444i)T + (18.6 - 3.70i)T^{2} \)
23 \( 1 + (4.42 - 0.879i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-8.50 - 2.57i)T + (24.1 + 16.1i)T^{2} \)
31 \( 1 + (-3.76 - 3.76i)T + 31iT^{2} \)
37 \( 1 + (-0.754 + 7.66i)T + (-36.2 - 7.21i)T^{2} \)
41 \( 1 + (1.08 + 5.43i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (4.10 - 7.67i)T + (-23.8 - 35.7i)T^{2} \)
47 \( 1 + (-0.141 - 0.340i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (0.564 - 0.171i)T + (44.0 - 29.4i)T^{2} \)
59 \( 1 + (-1.78 - 2.17i)T + (-11.5 + 57.8i)T^{2} \)
61 \( 1 + (10.9 - 5.82i)T + (33.8 - 50.7i)T^{2} \)
67 \( 1 + (-11.1 + 5.93i)T + (37.2 - 55.7i)T^{2} \)
71 \( 1 + (-5.71 - 8.55i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-10.3 - 6.91i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (1.93 - 4.65i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (0.186 + 1.89i)T + (-81.4 + 16.1i)T^{2} \)
89 \( 1 + (11.7 + 2.33i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (-10.7 - 10.7i)T + 97iT^{2} \)
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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

−10.84726444497284498573883322851, −10.17699657452373659320493109056, −9.453517423129111182491220166825, −8.610075733486348709269105349841, −8.015723496400913743665481222637, −6.46313799224952507214198826532, −5.16254374209734027325056369623, −4.16331546175846283711175831168, −2.49148536021749844092871418308, −1.27552803602075719862258905660, 1.81047059808323266468139575615, 2.61024207052274664320324896366, 5.02163561241987878962547801131, 6.19237031985348457124194402135, 6.61326554406556897738741550488, 7.84051981274003128735293525799, 8.523412590101742861783892942391, 9.868186760680184181089771692031, 10.13482043515134084253273942119, 11.20012417847496837947225096107

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