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Properties

Label 2-384-12.11-c5-0-44
Degree $2$
Conductor $384$
Sign $0.107 - 0.994i$
Analytic cond. $61.5873$
Root an. cond. $7.84776$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67 + 15.4i)3-s + 6.76i·5-s + 132. i·7-s + (−237. − 51.9i)9-s + 687.·11-s + 609.·13-s + (−104. − 11.3i)15-s − 550. i·17-s + 232. i·19-s + (−2.05e3 − 222. i)21-s + 4.48e3·23-s + 3.07e3·25-s + (1.20e3 − 3.59e3i)27-s − 2.04e3i·29-s − 6.67e3i·31-s + ⋯
L(s)  = 1  + (−0.107 + 0.994i)3-s + 0.121i·5-s + 1.02i·7-s + (−0.976 − 0.213i)9-s + 1.71·11-s + 1.00·13-s + (−0.120 − 0.0130i)15-s − 0.462i·17-s + 0.147i·19-s + (−1.01 − 0.109i)21-s + 1.76·23-s + 0.985·25-s + (0.317 − 0.948i)27-s − 0.451i·29-s − 1.24i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.107 - 0.994i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ 0.107 - 0.994i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.532750716\)
\(L(\frac12)\) \(\approx\) \(2.532750716\)
\(L(\frac{7}{2})\) 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 \( 1 + (1.67 - 15.4i)T \)
good5 \( 1 - 6.76iT - 3.12e3T^{2} \)
7 \( 1 - 132. iT - 1.68e4T^{2} \)
11 \( 1 - 687.T + 1.61e5T^{2} \)
13 \( 1 - 609.T + 3.71e5T^{2} \)
17 \( 1 + 550. iT - 1.41e6T^{2} \)
19 \( 1 - 232. iT - 2.47e6T^{2} \)
23 \( 1 - 4.48e3T + 6.43e6T^{2} \)
29 \( 1 + 2.04e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.67e3iT - 2.86e7T^{2} \)
37 \( 1 - 2.60e3T + 6.93e7T^{2} \)
41 \( 1 - 1.27e4iT - 1.15e8T^{2} \)
43 \( 1 - 9.78e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.69e4T + 2.29e8T^{2} \)
53 \( 1 + 2.24e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.61e4T + 7.14e8T^{2} \)
61 \( 1 + 8.67e3T + 8.44e8T^{2} \)
67 \( 1 + 5.33e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.50e4T + 1.80e9T^{2} \)
73 \( 1 - 7.81e3T + 2.07e9T^{2} \)
79 \( 1 - 6.49e3iT - 3.07e9T^{2} \)
83 \( 1 + 1.01e5T + 3.93e9T^{2} \)
89 \( 1 - 5.92e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.46e5T + 8.58e9T^{2} \)
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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

−10.94922911667408996377285820624, −9.484368183222799562270303258090, −9.180570460781260819830160742563, −8.298273996141812699228124257508, −6.66775811778503541086518921067, −5.92399734252020101243867645673, −4.81771288816343790285743816774, −3.76692005108316311779992208300, −2.73787440781460201942415187846, −1.03153469004382624970196936962, 0.906454383861760824347509646514, 1.38692197703657525244496157225, 3.18148976443351863026069888272, 4.24793931520481277562569680177, 5.67334843087900683596970220635, 6.87924574695724176229745810448, 7.06381429616755464606846031426, 8.600722872555442432698111289172, 9.028520054133536452718862131590, 10.64995156777424829702267374999

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