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Properties

Label 2-384-12.11-c5-0-41
Degree $2$
Conductor $384$
Sign $0.969 - 0.244i$
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  + (15.1 − 3.81i)3-s + 40.5i·5-s − 81.4i·7-s + (213. − 115. i)9-s − 499.·11-s + 262.·13-s + (154. + 613. i)15-s + 565. i·17-s + 2.13e3i·19-s + (−310. − 1.23e3i)21-s + 3.65e3·23-s + 1.47e3·25-s + (2.79e3 − 2.55e3i)27-s − 7.31e3i·29-s + 2.21e3i·31-s + ⋯
L(s)  = 1  + (0.969 − 0.244i)3-s + 0.725i·5-s − 0.628i·7-s + (0.880 − 0.474i)9-s − 1.24·11-s + 0.430·13-s + (0.177 + 0.703i)15-s + 0.474i·17-s + 1.35i·19-s + (−0.153 − 0.609i)21-s + 1.43·23-s + 0.473·25-s + (0.737 − 0.675i)27-s − 1.61i·29-s + 0.414i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.969 - 0.244i$
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: \((2,\ 384,\ (\ :5/2),\ 0.969 - 0.244i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.099920502\)
\(L(\frac12)\) \(\approx\) \(3.099920502\)
\(L(\frac{7}{2})\) 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 \( 1 + (-15.1 + 3.81i)T \)
good5 \( 1 - 40.5iT - 3.12e3T^{2} \)
7 \( 1 + 81.4iT - 1.68e4T^{2} \)
11 \( 1 + 499.T + 1.61e5T^{2} \)
13 \( 1 - 262.T + 3.71e5T^{2} \)
17 \( 1 - 565. iT - 1.41e6T^{2} \)
19 \( 1 - 2.13e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.65e3T + 6.43e6T^{2} \)
29 \( 1 + 7.31e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.21e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.00e4T + 6.93e7T^{2} \)
41 \( 1 - 2.86e3iT - 1.15e8T^{2} \)
43 \( 1 - 8.28e3iT - 1.47e8T^{2} \)
47 \( 1 - 3.92e3T + 2.29e8T^{2} \)
53 \( 1 - 1.49e3iT - 4.18e8T^{2} \)
59 \( 1 - 6.36e3T + 7.14e8T^{2} \)
61 \( 1 + 5.09e3T + 8.44e8T^{2} \)
67 \( 1 - 5.87e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.46e4T + 1.80e9T^{2} \)
73 \( 1 - 3.48e4T + 2.07e9T^{2} \)
79 \( 1 - 3.71e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.77e4T + 3.93e9T^{2} \)
89 \( 1 + 3.84e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.48e5T + 8.58e9T^{2} \)
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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

−10.41824541003111637598143181845, −9.787465942769686640275729168209, −8.496567016499123095730401255628, −7.78839708100623558197023595869, −7.01787958138300809722371917109, −5.92102447601189612977793115915, −4.38008379906324371013469083612, −3.31658426021560419944436262114, −2.42612369720818147721089498421, −1.01485808430919264534849035745, 0.838535927572167531915751277619, 2.35815656321115704017013653963, 3.18190677881082605629141895052, 4.72636950320948414565331106629, 5.28085882419389468799672956313, 6.92521194374732552388469823199, 7.86734349739089931995204039334, 9.008553061439166705018166326109, 9.037972212034124270081317020174, 10.44164928783662107502662669216

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