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Properties

Label 1-48e2-2304.715-r1-0-0
Degree $1$
Conductor $2304$
Sign $-0.192 - 0.981i$
Analytic cond. $247.599$
Root an. cond. $247.599$
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  + (−0.0327 − 0.999i)5-s + (−0.751 − 0.659i)7-s + (−0.986 + 0.162i)11-s + (−0.528 + 0.849i)13-s + (0.382 + 0.923i)17-s + (−0.290 + 0.956i)19-s + (−0.0654 + 0.997i)23-s + (−0.997 + 0.0654i)25-s + (0.935 − 0.352i)29-s + (0.258 − 0.965i)31-s + (−0.634 + 0.773i)35-s + (0.956 − 0.290i)37-s + (−0.997 − 0.0654i)41-s + (−0.812 − 0.582i)43-s + (−0.130 + 0.991i)47-s + ⋯
L(s)  = 1  + (−0.0327 − 0.999i)5-s + (−0.751 − 0.659i)7-s + (−0.986 + 0.162i)11-s + (−0.528 + 0.849i)13-s + (0.382 + 0.923i)17-s + (−0.290 + 0.956i)19-s + (−0.0654 + 0.997i)23-s + (−0.997 + 0.0654i)25-s + (0.935 − 0.352i)29-s + (0.258 − 0.965i)31-s + (−0.634 + 0.773i)35-s + (0.956 − 0.290i)37-s + (−0.997 − 0.0654i)41-s + (−0.812 − 0.582i)43-s + (−0.130 + 0.991i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.192 - 0.981i$
Analytic conductor: \(247.599\)
Root analytic conductor: \(247.599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (715, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2304,\ (1:\ ),\ -0.192 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4754361828 - 0.5777125327i\)
\(L(\frac12)\) \(\approx\) \(0.4754361828 - 0.5777125327i\)
\(L(1)\) \(\approx\) \(0.7826690468 - 0.1039168425i\)
\(L(1)\) \(\approx\) \(0.7826690468 - 0.1039168425i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.0327 - 0.999i)T \)
7 \( 1 + (-0.751 - 0.659i)T \)
11 \( 1 + (-0.986 + 0.162i)T \)
13 \( 1 + (-0.528 + 0.849i)T \)
17 \( 1 + (0.382 + 0.923i)T \)
19 \( 1 + (-0.290 + 0.956i)T \)
23 \( 1 + (-0.0654 + 0.997i)T \)
29 \( 1 + (0.935 - 0.352i)T \)
31 \( 1 + (0.258 - 0.965i)T \)
37 \( 1 + (0.956 - 0.290i)T \)
41 \( 1 + (-0.997 - 0.0654i)T \)
43 \( 1 + (-0.812 - 0.582i)T \)
47 \( 1 + (-0.130 + 0.991i)T \)
53 \( 1 + (-0.773 + 0.634i)T \)
59 \( 1 + (-0.999 + 0.0327i)T \)
61 \( 1 + (0.412 - 0.910i)T \)
67 \( 1 + (0.582 + 0.812i)T \)
71 \( 1 + (-0.980 - 0.195i)T \)
73 \( 1 + (0.195 + 0.980i)T \)
79 \( 1 + (0.793 - 0.608i)T \)
83 \( 1 + (0.729 - 0.683i)T \)
89 \( 1 + (0.831 - 0.555i)T \)
97 \( 1 + (-0.965 + 0.258i)T \)
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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

−19.58256058425071252221431541173, −18.87276070145400321549218587354, −18.1383211969800638233516861342, −17.906951097739866426195540111283, −16.650686063962171516398956155523, −15.96649934299866282279802487530, −15.28879751464004323913594649917, −14.813131234412195111785384431978, −13.82859273531070363410637722095, −13.174852943875596031080304725315, −12.38449027885187249105767120694, −11.70047976029395755409083713019, −10.725478225932634298325224054164, −10.21070489631436553225051030164, −9.54352925078725810657341752025, −8.50077383841398073252061221538, −7.81241178830193534865980464599, −6.83859065824091107858740993264, −6.407543336861419683320105242583, −5.33033602828995257349303326897, −4.765684044595500773511637002822, −3.22278382887330589803191178437, −2.92524415589383820477717881510, −2.25059757850865208394534088388, −0.5773444692975266963492730590, 0.20253646603088353596154064050, 1.300934614836915072172414976094, 2.15327185558276223425594016731, 3.33976630175749243002537930793, 4.14719846896304450473453451463, 4.80978752086583596075650134905, 5.79151914744918002121694996195, 6.4390988004485116825254466392, 7.6221365362233124334931336870, 7.97688267771303323643776620587, 8.993269725694910167307368969918, 9.871073569843448269326441969917, 10.16274400414554433475200304986, 11.288071738516098864709862350444, 12.18241104788153136578046755015, 12.7257029677903607324296812062, 13.39147698743095993971673975126, 14.0221398448878629672023084506, 15.0478492731013442109706790289, 15.811409291239318696544449898863, 16.44367777188101309923003480468, 17.04174999887847750523416738242, 17.55817221891007666567631124013, 18.85513774260188329389771958096, 19.14799479818384294344172952738

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