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Properties

Label 2-384-24.11-c3-0-7
Degree $2$
Conductor $384$
Sign $-0.290 - 0.956i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.44 + 4.58i)3-s − 2.31·5-s − 29.6i·7-s + (−15 − 22.4i)9-s + 22.8i·11-s + 8.66i·13-s + (5.66 − 10.6i)15-s + 93.3i·17-s + 85.4·19-s + (135. + 72.6i)21-s + 116.·23-s − 119.·25-s + (139. − 13.7i)27-s − 103.·29-s + 10.6i·31-s + ⋯
L(s)  = 1  + (−0.471 + 0.881i)3-s − 0.207·5-s − 1.60i·7-s + (−0.555 − 0.831i)9-s + 0.625i·11-s + 0.184i·13-s + (0.0975 − 0.182i)15-s + 1.33i·17-s + 1.03·19-s + (1.41 + 0.755i)21-s + 1.05·23-s − 0.957·25-s + (0.995 − 0.0979i)27-s − 0.662·29-s + 0.0614i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.290 - 0.956i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.290 - 0.956i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.045719097\)
\(L(\frac12)\) \(\approx\) \(1.045719097\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.44 - 4.58i)T \)
good5 \( 1 + 2.31T + 125T^{2} \)
7 \( 1 + 29.6iT - 343T^{2} \)
11 \( 1 - 22.8iT - 1.33e3T^{2} \)
13 \( 1 - 8.66iT - 2.19e3T^{2} \)
17 \( 1 - 93.3iT - 4.91e3T^{2} \)
19 \( 1 - 85.4T + 6.85e3T^{2} \)
23 \( 1 - 116.T + 1.21e4T^{2} \)
29 \( 1 + 103.T + 2.43e4T^{2} \)
31 \( 1 - 10.6iT - 2.97e4T^{2} \)
37 \( 1 - 380. iT - 5.06e4T^{2} \)
41 \( 1 - 257. iT - 6.89e4T^{2} \)
43 \( 1 + 359.T + 7.95e4T^{2} \)
47 \( 1 + 293.T + 1.03e5T^{2} \)
53 \( 1 - 679.T + 1.48e5T^{2} \)
59 \( 1 - 45.8iT - 2.05e5T^{2} \)
61 \( 1 - 595. iT - 2.26e5T^{2} \)
67 \( 1 - 206.T + 3.00e5T^{2} \)
71 \( 1 - 996.T + 3.57e5T^{2} \)
73 \( 1 + 593.T + 3.89e5T^{2} \)
79 \( 1 - 910. iT - 4.93e5T^{2} \)
83 \( 1 - 332. iT - 5.71e5T^{2} \)
89 \( 1 + 821. iT - 7.04e5T^{2} \)
97 \( 1 - 420.T + 9.12e5T^{2} \)
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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

−11.12490987203404949015041961641, −10.13961949936861558525427209184, −9.799660669341962611255292407415, −8.433065224697915467312746124820, −7.35356791090999351712919537949, −6.46609948058308520088963008769, −5.13514648531272822878453173201, −4.19486153067814323169476375310, −3.42543556098329336507812365572, −1.18551483360187471304988402036, 0.42779299956063003281643348859, 2.09187824509514195469510534669, 3.17641306744327617180398276441, 5.25321819802951563985873285120, 5.62194567528563450031084560443, 6.85785440811193895837761307691, 7.77421222434977744817878065656, 8.754916080252737357594828354750, 9.532363971773938324467047827046, 11.05495045417006206688760147303

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