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Properties

Label 1-381-381.377-r0-0-0
Degree $1$
Conductor $381$
Sign $0.805 - 0.592i$
Analytic cond. $1.76935$
Root an. cond. $1.76935$
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  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + (0.900 − 0.433i)7-s + (−0.623 + 0.781i)8-s + (0.222 + 0.974i)10-s + (0.222 + 0.974i)11-s + (−0.900 + 0.433i)13-s + (−0.222 − 0.974i)14-s + (0.623 + 0.781i)16-s + (0.900 − 0.433i)17-s + 19-s + 20-s + 22-s + (−0.222 + 0.974i)23-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + (0.900 − 0.433i)7-s + (−0.623 + 0.781i)8-s + (0.222 + 0.974i)10-s + (0.222 + 0.974i)11-s + (−0.900 + 0.433i)13-s + (−0.222 − 0.974i)14-s + (0.623 + 0.781i)16-s + (0.900 − 0.433i)17-s + 19-s + 20-s + 22-s + (−0.222 + 0.974i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(381\)    =    \(3 \cdot 127\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(1.76935\)
Root analytic conductor: \(1.76935\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{381} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 381,\ (0:\ ),\ 0.805 - 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.126830421 - 0.3694445332i\)
\(L(\frac12)\) \(\approx\) \(1.126830421 - 0.3694445332i\)
\(L(1)\) \(\approx\) \(0.9683714958 - 0.3554103906i\)
\(L(1)\) \(\approx\) \(0.9683714958 - 0.3554103906i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (0.900 - 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 + T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + (-0.623 - 0.781i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−24.581278107486445703283840101694, −23.94045337324011725276160227070, −23.13647350123562085371229366383, −22.11180706928027137456541979696, −21.39815256042556233752139140153, −20.29767614266204805179456495781, −19.19070326489440031379909784506, −18.38135272589680073373976377356, −17.39808606983356900129115321359, −16.4841973270366163406502842432, −15.87520622707523022205491777011, −14.7108175952445778829570057053, −14.39378743304914685669373252113, −13.00983085968885963875751128504, −12.14389848397243344792686585485, −11.376912452457428977477032954856, −9.870796208772597634858639164216, −8.6378306665584831222735457078, −8.07882473272690887522415696388, −7.31090754385368815205100759777, −5.875708145010753470064694817772, −5.090572056260930185931508280682, −4.14426379919825066750271694185, −2.98724641602235194372247123420, −0.85749105889228285381487861525, 1.1576091691994638770838416251, 2.42447794169163036495337611841, 3.638449688092887573448199722393, 4.49875759696393233832072000098, 5.37002521504899229969102822544, 7.23339986692586208716466951202, 7.77335839040859235495109037238, 9.213710744795164820755885861982, 10.08064462658231099413471036202, 11.0781307109182623541096199296, 11.869413717112171281337143569304, 12.37949937411977212992103828322, 13.86359430530540582459712220462, 14.49963111419731969668465382134, 15.22813923179707579149482313062, 16.62272151080275195909436500691, 17.7559416402867516392773730777, 18.376023360140895114652541709913, 19.46053163843219409900454683289, 20.080924763419151234053928027960, 20.798037482823498800937628947566, 21.89974195649780203815320570485, 22.586211381675283187954339209123, 23.58750658966043081792045124311, 23.93118823512458678534408569359

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