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Properties

Label 1-384-384.59-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: $\chi_{384} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (0:\ ),\ 0.989 - 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.677035102 - 0.1237055409i\)
\(L(\frac12)\) \(\approx\) \(1.677035102 - 0.1237055409i\)
\(L(1)\) \(\approx\) \(1.334552535 - 0.04939433649i\)
\(L(1)\) \(\approx\) \(1.334552535 - 0.04939433649i\)

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

−24.75475203910188224588059205377, −23.84116193095430658060998487236, −22.606440890821668023221602407200, −21.80230035425462446900643631895, −21.18843864946781327765989061904, −20.36310600697115906716514644451, −19.15321930915499247004115076994, −18.202920721798044140795793429591, −17.76745610442732732620746318537, −16.43556985938528152186989852150, −15.92667280882835303990529485207, −14.46464020333052419518549545681, −14.006754974083798226626249025410, −12.93399897332746300087783131574, −11.95022259428753917210655603400, −11.02794519085384374223586610297, −9.97370559279492480457772091031, −8.779818576359700817303872550046, −8.511095349642666350459667877305, −6.71195128991265579424087095141, −5.932502172650638504492895285720, −5.105513448110488967105798350799, −3.70216838237664380365487524068, −2.37450801964620767045326140, −1.37912851286621529313223745649, 1.28251711843656962119282885676, 2.28479381134179696274472999858, 3.830327521430855064158027111328, 4.72362070138373878669149753554, 6.10275829290797136565161669525, 6.778401505763132776670157506170, 7.97176481649235337418117810461, 9.09266753027972009212405544718, 10.06978749246160415364688919814, 10.79400908490481587520589379393, 11.81237667427665818417359535819, 13.27988513671878082968765476204, 13.556365290723491964190688081183, 14.675509641287711075733836512676, 15.52426160176351401699909286878, 16.8317279374829212976299356655, 17.54553347168992140849983859967, 18.01418755252915464737739408228, 19.34777173954271994870958744439, 20.282059723180731424447195145722, 20.91254312681455083287550023639, 21.946361290457041655431605942878, 22.65867895678777665670000685846, 23.70656377773130412775351432678, 24.41630259923152439557212741194

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