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Properties

Label 2-384-48.11-c3-0-8
Degree $2$
Conductor $384$
Sign $-0.147 - 0.989i$
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  + (−5.19 + 0.151i)3-s + (−4.66 + 4.66i)5-s + 0.405·7-s + (26.9 − 1.56i)9-s + (5.82 + 5.82i)11-s + (35.2 − 35.2i)13-s + (23.5 − 24.9i)15-s − 49.3i·17-s + (−108. − 108. i)19-s + (−2.10 + 0.0612i)21-s + 130. i·23-s + 81.4i·25-s + (−139. + 12.2i)27-s + (172. + 172. i)29-s − 36.1i·31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0290i)3-s + (−0.417 + 0.417i)5-s + 0.0219·7-s + (0.998 − 0.0581i)9-s + (0.159 + 0.159i)11-s + (0.751 − 0.751i)13-s + (0.405 − 0.429i)15-s − 0.703i·17-s + (−1.30 − 1.30i)19-s + (−0.0219 + 0.000636i)21-s + 1.18i·23-s + 0.651i·25-s + (−0.996 + 0.0870i)27-s + (1.10 + 1.10i)29-s − 0.209i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.147 - 0.989i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.147 - 0.989i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8124088479\)
\(L(\frac12)\) \(\approx\) \(0.8124088479\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.19 - 0.151i)T \)
good5 \( 1 + (4.66 - 4.66i)T - 125iT^{2} \)
7 \( 1 - 0.405T + 343T^{2} \)
11 \( 1 + (-5.82 - 5.82i)T + 1.33e3iT^{2} \)
13 \( 1 + (-35.2 + 35.2i)T - 2.19e3iT^{2} \)
17 \( 1 + 49.3iT - 4.91e3T^{2} \)
19 \( 1 + (108. + 108. i)T + 6.85e3iT^{2} \)
23 \( 1 - 130. iT - 1.21e4T^{2} \)
29 \( 1 + (-172. - 172. i)T + 2.43e4iT^{2} \)
31 \( 1 + 36.1iT - 2.97e4T^{2} \)
37 \( 1 + (-257. - 257. i)T + 5.06e4iT^{2} \)
41 \( 1 + 5.87T + 6.89e4T^{2} \)
43 \( 1 + (170. - 170. i)T - 7.95e4iT^{2} \)
47 \( 1 + 181.T + 1.03e5T^{2} \)
53 \( 1 + (-148. + 148. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-567. - 567. i)T + 2.05e5iT^{2} \)
61 \( 1 + (481. - 481. i)T - 2.26e5iT^{2} \)
67 \( 1 + (296. + 296. i)T + 3.00e5iT^{2} \)
71 \( 1 - 533. iT - 3.57e5T^{2} \)
73 \( 1 - 178. iT - 3.89e5T^{2} \)
79 \( 1 - 528. iT - 4.93e5T^{2} \)
83 \( 1 + (713. - 713. i)T - 5.71e5iT^{2} \)
89 \( 1 - 204.T + 7.04e5T^{2} \)
97 \( 1 - 275.T + 9.12e5T^{2} \)
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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

−11.24171263066664627016703754629, −10.49508101252584084974226818296, −9.503513877789706638915693528591, −8.315001402102213121633187153249, −7.16669890715541756083779330675, −6.48291584959530153478003184977, −5.34084235872294679715070870535, −4.35890733707210098843641605749, −3.03446267870067554043365940418, −1.12887065471995574791650905232, 0.37448085133898057247834329785, 1.82565193161419379293945733872, 3.96685738704874826257907208464, 4.58180707858343526754378015411, 6.09825823775371832871645725009, 6.47162944656888461858446985399, 7.976522160675951268646321658663, 8.667327045177697917994262970664, 10.01726275025567506891505497425, 10.72903583874258901964054065874

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