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Properties

Label 1-384-384.179-r0-0-0
Degree $1$
Conductor $384$
Sign $-0.989 - 0.146i$
Analytic cond. $1.78328$
Root an. cond. $1.78328$
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  + (−0.980 + 0.195i)5-s + (0.382 + 0.923i)7-s + (−0.555 + 0.831i)11-s + (−0.980 − 0.195i)13-s + (−0.707 − 0.707i)17-s + (0.195 − 0.980i)19-s + (−0.923 − 0.382i)23-s + (0.923 − 0.382i)25-s + (−0.555 − 0.831i)29-s + i·31-s + (−0.555 − 0.831i)35-s + (0.195 + 0.980i)37-s + (−0.923 − 0.382i)41-s + (−0.831 − 0.555i)43-s + (−0.707 − 0.707i)47-s + ⋯
L(s)  = 1  + (−0.980 + 0.195i)5-s + (0.382 + 0.923i)7-s + (−0.555 + 0.831i)11-s + (−0.980 − 0.195i)13-s + (−0.707 − 0.707i)17-s + (0.195 − 0.980i)19-s + (−0.923 − 0.382i)23-s + (0.923 − 0.382i)25-s + (−0.555 − 0.831i)29-s + i·31-s + (−0.555 − 0.831i)35-s + (0.195 + 0.980i)37-s + (−0.923 − 0.382i)41-s + (−0.831 − 0.555i)43-s + (−0.707 − 0.707i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.989 - 0.146i$
Analytic conductor: \(1.78328\)
Root analytic conductor: \(1.78328\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (0:\ ),\ -0.989 - 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009017841322 + 0.1222518920i\)
\(L(\frac12)\) \(\approx\) \(0.009017841322 + 0.1222518920i\)
\(L(1)\) \(\approx\) \(0.6373423386 + 0.1057861385i\)
\(L(1)\) \(\approx\) \(0.6373423386 + 0.1057861385i\)

Euler product

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

−24.03623919235624776035221405760, −23.39716383981898704526106923229, −22.37572381745390134249939708178, −21.44079281643751908843134078630, −20.38035504131096329229201486537, −19.778610416362195730863307158626, −18.939193148733998936426581337446, −17.91676282379421812783741202703, −16.778606079637734783268074668280, −16.281606876865908310349550535798, −15.14145042192195026367307297431, −14.335957934749934229768119167832, −13.30542765013195688453688770870, −12.35431798938123408065016381827, −11.33765461249175181994840998266, −10.66011425924478008462876356468, −9.519852077082537960041256606591, −8.0868278614355522335634298791, −7.79254928355289335663062529565, −6.55608194866127383668661020529, −5.19863537198080534432078213972, −4.1845365062775470646948844265, −3.35404877765823764635753463687, −1.71799940169238658444260744682, −0.06918442042085754519286271868, 2.11984022368195714336568796052, 3.00314994549062163604666674824, 4.55012389806424919646528915509, 5.11934614995715461715426951352, 6.68014646092262827860360843083, 7.55734907909982105919733729509, 8.42733013374864112244547231014, 9.48951227505392010173569989607, 10.57802977905635188453266659254, 11.74917688057395723147784593483, 12.126146585119183451390758252649, 13.281353721737613928355995994800, 14.56058664594862141528559883495, 15.38309189332935741527058080592, 15.7559804981222909582612338486, 17.13977499692425702945630035780, 18.116446093939533463130592238569, 18.71792539111925207242756472533, 19.907215624863495053969296909565, 20.34002987708257489616425657088, 21.69671291360427560702262408530, 22.325534623588672825376464730140, 23.22189068582058319759015134573, 24.18939064642502806446784140490, 24.76857837712468172569705610472

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