Properties

Label 2-384-16.11-c6-0-13
Degree $2$
Conductor $384$
Sign $0.856 + 0.516i$
Analytic cond. $88.3407$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.0 − 11.0i)3-s + (−156. − 156. i)5-s + 23.3·7-s + 242. i·9-s + (−1.21e3 + 1.21e3i)11-s + (1.85e3 − 1.85e3i)13-s + 3.44e3i·15-s − 445.·17-s + (−5.08e3 − 5.08e3i)19-s + (−257. − 257. i)21-s − 5.62e3·23-s + 3.32e4i·25-s + (2.67e3 − 2.67e3i)27-s + (−1.65e4 + 1.65e4i)29-s + 4.88e4i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−1.25 − 1.25i)5-s + 0.0681·7-s + 0.333i·9-s + (−0.910 + 0.910i)11-s + (0.842 − 0.842i)13-s + 1.02i·15-s − 0.0907·17-s + (−0.742 − 0.742i)19-s + (−0.0278 − 0.0278i)21-s − 0.462·23-s + 2.12i·25-s + (0.136 − 0.136i)27-s + (−0.678 + 0.678i)29-s + 1.63i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.856 + 0.516i$
Analytic conductor: \(88.3407\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3),\ 0.856 + 0.516i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.7526894830\)
\(L(\frac12)\) \(\approx\) \(0.7526894830\)
\(L(4)\) 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 \( 1 + (11.0 + 11.0i)T \)
good5 \( 1 + (156. + 156. i)T + 1.56e4iT^{2} \)
7 \( 1 - 23.3T + 1.17e5T^{2} \)
11 \( 1 + (1.21e3 - 1.21e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (-1.85e3 + 1.85e3i)T - 4.82e6iT^{2} \)
17 \( 1 + 445.T + 2.41e7T^{2} \)
19 \( 1 + (5.08e3 + 5.08e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 5.62e3T + 1.48e8T^{2} \)
29 \( 1 + (1.65e4 - 1.65e4i)T - 5.94e8iT^{2} \)
31 \( 1 - 4.88e4iT - 8.87e8T^{2} \)
37 \( 1 + (-5.29e4 - 5.29e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 9.48e3iT - 4.75e9T^{2} \)
43 \( 1 + (3.22e4 - 3.22e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 6.36e4iT - 1.07e10T^{2} \)
53 \( 1 + (1.26e5 + 1.26e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (1.15e5 - 1.15e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (-1.82e5 + 1.82e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (9.34e4 + 9.34e4i)T + 9.04e10iT^{2} \)
71 \( 1 - 5.32e5T + 1.28e11T^{2} \)
73 \( 1 + 4.93e5iT - 1.51e11T^{2} \)
79 \( 1 + 7.44e5iT - 2.43e11T^{2} \)
83 \( 1 + (-3.87e5 - 3.87e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 1.17e6iT - 4.96e11T^{2} \)
97 \( 1 - 1.27e6T + 8.32e11T^{2} \)
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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

−10.46881735282007015448278615415, −9.132567824305551876996946217369, −8.164751925842336929501761793724, −7.73200616230996393011773793753, −6.51869551751757557869706781113, −5.12455872145475036406610078758, −4.62872959113921222598692475185, −3.29615587971272760311306884676, −1.61966657545231017071782194270, −0.49801451223786250983514907415, 0.37682325623158591979960417876, 2.37405828582863583839703240143, 3.64935715302632394693212315408, 4.17259749571062021143950092032, 5.79699258542724401346716081788, 6.53541191914715146349014833442, 7.72257813387958671081814216008, 8.319073336957396571316978253151, 9.686782740367277236301889784155, 10.78188287888230733783289427968

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