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Properties

Label 1-384-384.53-r1-0-0
Degree $1$
Conductor $384$
Sign $-0.242 + 0.970i$
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  + (−0.555 − 0.831i)5-s + (−0.923 − 0.382i)7-s + (0.195 − 0.980i)11-s + (−0.555 + 0.831i)13-s + (0.707 − 0.707i)17-s + (−0.831 − 0.555i)19-s + (−0.382 − 0.923i)23-s + (−0.382 + 0.923i)25-s + (−0.195 − 0.980i)29-s + i·31-s + (0.195 + 0.980i)35-s + (0.831 − 0.555i)37-s + (0.382 + 0.923i)41-s + (−0.980 − 0.195i)43-s + (−0.707 + 0.707i)47-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)5-s + (−0.923 − 0.382i)7-s + (0.195 − 0.980i)11-s + (−0.555 + 0.831i)13-s + (0.707 − 0.707i)17-s + (−0.831 − 0.555i)19-s + (−0.382 − 0.923i)23-s + (−0.382 + 0.923i)25-s + (−0.195 − 0.980i)29-s + i·31-s + (0.195 + 0.980i)35-s + (0.831 − 0.555i)37-s + (0.382 + 0.923i)41-s + (−0.980 − 0.195i)43-s + (−0.707 + 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.242 + 0.970i$
Analytic conductor: \(41.2665\)
Root analytic conductor: \(41.2665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (1:\ ),\ -0.242 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07862628854 + 0.1007502778i\)
\(L(\frac12)\) \(\approx\) \(0.07862628854 + 0.1007502778i\)
\(L(1)\) \(\approx\) \(0.6940245302 - 0.1827811781i\)
\(L(1)\) \(\approx\) \(0.6940245302 - 0.1827811781i\)

Euler product

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

−23.8169545891042064010923485680, −23.08702035817385466277995359061, −22.368321380825607864345243908620, −21.710519267478123782588118548935, −20.36954411111737408472612495593, −19.55609430257398857632128324497, −18.92134356480105521723752231357, −17.97924545517752800804988216369, −17.01362761375406531455528780045, −15.951418295387993125582559318092, −14.99550178328908729647488870520, −14.65284731475236040304507851765, −13.07998318117400439243364639048, −12.41985829989411598803044961509, −11.5004754104154078069182542554, −10.19897711963733108859903493112, −9.814668316462546033644168531777, −8.31921285022808510654507210537, −7.403538740016460790440439896849, −6.5036906092018088582421305601, −5.48847314900607054165334019148, −3.99646668634753653379700891423, −3.17036071840640156542545654410, −1.9996490175961480654634875411, −0.04248444739311075056261525052, 0.9535189788979928598924825137, 2.68571826214489967725434254772, 3.86822363603317665276877147115, 4.730675271205801042218047816898, 6.037770706273497695341562959718, 7.00354575701904398472157391944, 8.12828503335331315358553723630, 9.06190324082268470311363826586, 9.87834012676972402901627702931, 11.14585827129637117452702025579, 12.02710891878187866550835555396, 12.88267265236453393598018084063, 13.74799461194719644475942469674, 14.75707643532375195231463750420, 16.11680177765794154039199269442, 16.39320511460302413010174582442, 17.22705446905042477785396041670, 18.70293259352325625540617626282, 19.37458494465386281546283945377, 20.01731953961317513584479572248, 21.090540299443085240652647813990, 21.87775205342188764700982409988, 22.99290331783853589425157056324, 23.64663645838024539456928188508, 24.51031857702268004363369660516

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