Properties

Degree $2$
Conductor $384$
Sign $-0.702 - 0.711i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 1.22i)3-s + (4.78 + 4.78i)5-s − 10.3·7-s + 2.99i·9-s + (0.526 − 0.526i)11-s + (−17.2 + 17.2i)13-s + 11.7i·15-s + 4.71·17-s + (2.53 + 2.53i)19-s + (−12.6 − 12.6i)21-s − 12.5·23-s + 20.8i·25-s + (−3.67 + 3.67i)27-s + (2.19 − 2.19i)29-s − 28.0i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.957 + 0.957i)5-s − 1.47·7-s + 0.333i·9-s + (0.0478 − 0.0478i)11-s + (−1.32 + 1.32i)13-s + 0.781i·15-s + 0.277·17-s + (0.133 + 0.133i)19-s + (−0.602 − 0.602i)21-s − 0.547·23-s + 0.834i·25-s + (−0.136 + 0.136i)27-s + (0.0757 − 0.0757i)29-s − 0.904i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.702 - 0.711i$
Motivic weight: \(2\)
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.702 - 0.711i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.543155 + 1.29894i\)
\(L(\frac12)\) \(\approx\) \(0.543155 + 1.29894i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.22 - 1.22i)T \)
good5 \( 1 + (-4.78 - 4.78i)T + 25iT^{2} \)
7 \( 1 + 10.3T + 49T^{2} \)
11 \( 1 + (-0.526 + 0.526i)T - 121iT^{2} \)
13 \( 1 + (17.2 - 17.2i)T - 169iT^{2} \)
17 \( 1 - 4.71T + 289T^{2} \)
19 \( 1 + (-2.53 - 2.53i)T + 361iT^{2} \)
23 \( 1 + 12.5T + 529T^{2} \)
29 \( 1 + (-2.19 + 2.19i)T - 841iT^{2} \)
31 \( 1 + 28.0iT - 961T^{2} \)
37 \( 1 + (-32.1 - 32.1i)T + 1.36e3iT^{2} \)
41 \( 1 - 23.1iT - 1.68e3T^{2} \)
43 \( 1 + (4.79 - 4.79i)T - 1.84e3iT^{2} \)
47 \( 1 - 39.0iT - 2.20e3T^{2} \)
53 \( 1 + (-27.9 - 27.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (79.8 - 79.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (-36.7 + 36.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (-10.9 - 10.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 52.6T + 5.04e3T^{2} \)
73 \( 1 - 67.8iT - 5.32e3T^{2} \)
79 \( 1 + 56.4iT - 6.24e3T^{2} \)
83 \( 1 + (-58.3 - 58.3i)T + 6.88e3iT^{2} \)
89 \( 1 + 131. iT - 7.92e3T^{2} \)
97 \( 1 - 60.9T + 9.40e3T^{2} \)
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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

−11.36822531052908815286483844928, −10.05208733679081152939697775508, −9.849990663536787443847673768096, −9.125727596348082348878346264219, −7.56102870471195729903776170195, −6.62215732347137006082491300964, −5.95007263336434608089707146864, −4.40851703799393869293277079588, −3.10227036384782985042444699202, −2.26582385691983653767905702343, 0.55162671110084080199726932682, 2.27248154052480975499635691539, 3.39865271005758223598699402528, 5.07954778127749834715387097072, 5.91517515166145522930977589988, 6.97547800586596217103764244073, 8.025475849319100876710978588795, 9.159045742366463811518668260741, 9.717750735865751877456299236110, 10.40357931211943092079787438066

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