Properties

Label 2-384-24.5-c2-0-11
Degree $2$
Conductor $384$
Sign $-0.333 - 0.942i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 2.82i)3-s + (−7.00 + 5.65i)9-s + 14·11-s + 33.9i·17-s + 16.9i·19-s − 25·25-s + (−23.0 − 14.1i)27-s + (14 + 39.5i)33-s + 67.8i·41-s − 84.8i·43-s − 49·49-s + (−96 + 33.9i)51-s + (−48 + 16.9i)57-s + 82·59-s + 118. i·67-s + ⋯
L(s)  = 1  + (0.333 + 0.942i)3-s + (−0.777 + 0.628i)9-s + 1.27·11-s + 1.99i·17-s + 0.893i·19-s − 25-s + (−0.851 − 0.523i)27-s + (0.424 + 1.19i)33-s + 1.65i·41-s − 1.97i·43-s − 0.999·49-s + (−1.88 + 0.665i)51-s + (−0.842 + 0.297i)57-s + 1.38·59-s + 1.77i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.333 - 0.942i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.333 - 0.942i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.996382 + 1.40909i\)
\(L(\frac12)\) \(\approx\) \(0.996382 + 1.40909i\)
\(L(2)\) 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 \( 1 + (-1 - 2.82i)T \)
good5 \( 1 + 25T^{2} \)
7 \( 1 + 49T^{2} \)
11 \( 1 - 14T + 121T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 - 33.9iT - 289T^{2} \)
19 \( 1 - 16.9iT - 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 - 67.8iT - 1.68e3T^{2} \)
43 \( 1 + 84.8iT - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 82T + 3.48e3T^{2} \)
61 \( 1 - 3.72e3T^{2} \)
67 \( 1 - 118. iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 142T + 5.32e3T^{2} \)
79 \( 1 + 6.24e3T^{2} \)
83 \( 1 - 158T + 6.88e3T^{2} \)
89 \( 1 + 101. iT - 7.92e3T^{2} \)
97 \( 1 + 94T + 9.40e3T^{2} \)
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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

−11.30055324486475863609284737102, −10.33138506087614028230797350441, −9.677001820301203809842832195341, −8.662613721105539014594154100082, −7.996752552438234226508357093334, −6.48160258826234289988760967008, −5.59102255421789313541249810055, −4.15431797147578082139720404995, −3.59643568049777889321244605669, −1.82411520959539345302716355820, 0.75509870246957107200116109617, 2.26753061296179472572078955387, 3.52472761402188817023236344179, 4.99198788769807268505217920982, 6.32963652806260335913268035065, 7.04323302708937753741280308066, 7.928489373145837548766333028808, 9.121667426761521819345436313131, 9.546970934611118927338874608792, 11.23527059280567985679956362223

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