Properties

Label 1-384-384.347-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  + (−0.195 + 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

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

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: $\chi_{384} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (0:\ ),\ -0.989 + 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03480619565 + 0.4718560821i\)
\(L(\frac12)\) \(\approx\) \(0.03480619565 + 0.4718560821i\)
\(L(1)\) \(\approx\) \(0.6798036605 + 0.2734134600i\)
\(L(1)\) \(\approx\) \(0.6798036605 + 0.2734134600i\)

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

−23.84301510583782034152381625934, −23.582148222705068252213044611044, −22.42860478852347832070651555684, −21.21854087124579494278291825073, −20.744395395874985771175465879722, −19.65398005988418931105935561121, −19.114114288646854629715749675477, −17.81640083849203382002585852594, −16.84390429765991513062674075707, −16.299522623584418099202040975805, −15.442283654113260161164556063129, −14.08069788345342348657988220127, −13.29952875128151036205479861341, −12.635148916365908812053286830027, −11.39724029870833221303843479651, −10.60504013809518671146202820295, −9.35942150871504013875684733485, −8.66699403818819708577769002007, −7.48446197692207170871286150425, −6.61622688134364213586911839257, −5.19562020094302442172021632772, −4.42550651794927174124287661050, −3.286724888467350029560936421441, −1.75483395475361653478307602505, −0.262888631514948601990646574474, 2.21723281626485873166462576125, 2.8730566888067999823782516402, 4.185136751207016613723355495227, 5.55039132245885544027213157965, 6.37924582915483121048106708763, 7.48333476720825012731468696615, 8.39061311532190730571126312575, 9.598371918997422495374785821519, 10.54949166572722613328688550217, 11.25284069980121046445783622784, 12.624761977053404597024783442297, 13.02616793527515453092054146994, 14.56950123280374572919748483210, 15.23567836865480175785597642692, 15.717570309956965771496122469516, 17.21141102951318739915108570937, 17.981088977874840370128532981668, 18.8703117686673875533326387090, 19.42255722122782546917967697561, 20.67418761003438697125410305189, 21.51532406091593409776923934380, 22.57746626053384892060367308368, 22.83602086524139489182817141377, 24.07241017659984896475282055760, 25.036469889850407203169541262799

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