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Properties

Label 1-384-384.365-r1-0-0
Degree $1$
Conductor $384$
Sign $0.146 - 0.989i$
Analytic cond. $41.2665$
Root an. cond. $41.2665$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.195 + 0.980i)5-s + (0.382 + 0.923i)7-s + (−0.831 − 0.555i)11-s + (0.195 − 0.980i)13-s + (−0.707 − 0.707i)17-s + (−0.980 − 0.195i)19-s + (−0.923 − 0.382i)23-s + (−0.923 + 0.382i)25-s + (0.831 − 0.555i)29-s − i·31-s + (−0.831 + 0.555i)35-s + (0.980 − 0.195i)37-s + (0.923 + 0.382i)41-s + (0.555 − 0.831i)43-s + (0.707 + 0.707i)47-s + ⋯
L(s)  = 1  + (0.195 + 0.980i)5-s + (0.382 + 0.923i)7-s + (−0.831 − 0.555i)11-s + (0.195 − 0.980i)13-s + (−0.707 − 0.707i)17-s + (−0.980 − 0.195i)19-s + (−0.923 − 0.382i)23-s + (−0.923 + 0.382i)25-s + (0.831 − 0.555i)29-s − i·31-s + (−0.831 + 0.555i)35-s + (0.980 − 0.195i)37-s + (0.923 + 0.382i)41-s + (0.555 − 0.831i)43-s + (0.707 + 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.146 - 0.989i$
Analytic conductor: \(41.2665\)
Root analytic conductor: \(41.2665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (1:\ ),\ 0.146 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7982915307 - 0.6886110101i\)
\(L(\frac12)\) \(\approx\) \(0.7982915307 - 0.6886110101i\)
\(L(1)\) \(\approx\) \(0.9492769335 + 0.03401531621i\)
\(L(1)\) \(\approx\) \(0.9492769335 + 0.03401531621i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.195 + 0.980i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (-0.831 - 0.555i)T \)
13 \( 1 + (0.195 - 0.980i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (-0.980 - 0.195i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (0.831 - 0.555i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.980 - 0.195i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (0.555 - 0.831i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.831 + 0.555i)T \)
59 \( 1 + (-0.195 - 0.980i)T \)
61 \( 1 + (-0.555 - 0.831i)T \)
67 \( 1 + (-0.555 - 0.831i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (-0.980 - 0.195i)T \)
89 \( 1 + (-0.923 + 0.382i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.194076701447965132291186000903, −23.772485339762514263077182948302, −23.04883770464213544696960824872, −21.4809909743681406473829647248, −21.16995843033946258180620800207, −20.06105881561358832130989918875, −19.5653150879972378457507461411, −18.117578107378715686126854314019, −17.44599085072081867068508092571, −16.563235696443215197934567598026, −15.841233547107755041127088470028, −14.63305972263894870212875844217, −13.66775185561460084266571537676, −12.939657978308379205695983287910, −12.03074108733584926406209024309, −10.82667000331874374378815012277, −10.05692090166896334346938875292, −8.87749007165944945227502020420, −8.09104803902728023863833983491, −7.01238887009485818906646771920, −5.85378810275008022285279883453, −4.56065881477534540029255471996, −4.1231952157142948480819446951, −2.21945659927037203995387731527, −1.22461241183957933962548057533, 0.29310534390922544889944033691, 2.32735817233445744090279513192, 2.790364494652196776777625063821, 4.326157590546893784434849688113, 5.66726268269438584227556133738, 6.26343845966069753112048036881, 7.65081710282482990517486565915, 8.4036674188776895595943910956, 9.59985954284195390772264957271, 10.69128907957211452987976680644, 11.2535673995225763994053884176, 12.44172843946852237031572056910, 13.44049087833037574650574632610, 14.34551962572179839461300863265, 15.370594813977334190197536727510, 15.76668722268293441333213554735, 17.27716522521937875718465728585, 18.20285184336749206991691618546, 18.54606994267184703532829197734, 19.63226085325118348645318101039, 20.774985039416156006968829355127, 21.59616704615042883151521816389, 22.28436916149462149315669211279, 23.12465660340969804116824576719, 24.1564379567855353108907934303

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