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Properties

Label 1-48e2-2304.691-r1-0-0
Degree $1$
Conductor $2304$
Sign $0.00272 + 0.999i$
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  + (0.412 − 0.910i)5-s + (0.997 + 0.0654i)7-s + (−0.528 + 0.849i)11-s + (0.812 + 0.582i)13-s + (−0.923 − 0.382i)17-s + (−0.634 − 0.773i)19-s + (−0.751 − 0.659i)23-s + (−0.659 − 0.751i)25-s + (0.0327 + 0.999i)29-s + (−0.258 − 0.965i)31-s + (0.471 − 0.881i)35-s + (0.773 + 0.634i)37-s + (−0.659 + 0.751i)41-s + (−0.227 − 0.973i)43-s + (−0.991 + 0.130i)47-s + ⋯
L(s)  = 1  + (0.412 − 0.910i)5-s + (0.997 + 0.0654i)7-s + (−0.528 + 0.849i)11-s + (0.812 + 0.582i)13-s + (−0.923 − 0.382i)17-s + (−0.634 − 0.773i)19-s + (−0.751 − 0.659i)23-s + (−0.659 − 0.751i)25-s + (0.0327 + 0.999i)29-s + (−0.258 − 0.965i)31-s + (0.471 − 0.881i)35-s + (0.773 + 0.634i)37-s + (−0.659 + 0.751i)41-s + (−0.227 − 0.973i)43-s + (−0.991 + 0.130i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00272 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00272 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.00272 + 0.999i$
Analytic conductor: \(247.599\)
Root analytic conductor: \(247.599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2304,\ (1:\ ),\ 0.00272 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9608369821 + 0.9582202721i\)
\(L(\frac12)\) \(\approx\) \(0.9608369821 + 0.9582202721i\)
\(L(1)\) \(\approx\) \(1.100201963 - 0.04835404813i\)
\(L(1)\) \(\approx\) \(1.100201963 - 0.04835404813i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.412 - 0.910i)T \)
7 \( 1 + (0.997 + 0.0654i)T \)
11 \( 1 + (-0.528 + 0.849i)T \)
13 \( 1 + (0.812 + 0.582i)T \)
17 \( 1 + (-0.923 - 0.382i)T \)
19 \( 1 + (-0.634 - 0.773i)T \)
23 \( 1 + (-0.751 - 0.659i)T \)
29 \( 1 + (0.0327 + 0.999i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (0.773 + 0.634i)T \)
41 \( 1 + (-0.659 + 0.751i)T \)
43 \( 1 + (-0.227 - 0.973i)T \)
47 \( 1 + (-0.991 + 0.130i)T \)
53 \( 1 + (-0.881 + 0.471i)T \)
59 \( 1 + (0.910 + 0.412i)T \)
61 \( 1 + (0.683 - 0.729i)T \)
67 \( 1 + (-0.973 - 0.227i)T \)
71 \( 1 + (0.831 + 0.555i)T \)
73 \( 1 + (0.555 + 0.831i)T \)
79 \( 1 + (-0.608 + 0.793i)T \)
83 \( 1 + (0.935 + 0.352i)T \)
89 \( 1 + (0.195 + 0.980i)T \)
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

−19.181115094493739590630364863392, −18.42394394789788163589318361101, −17.84570984303332839455831164048, −17.44326707251520734010776767914, −16.33047455632244090741629013558, −15.62167040372489426675475917561, −14.86781024801672561658571045755, −14.27184640100498046079984790442, −13.51156031579629381203504361710, −13.00765330361469261179299054788, −11.74989324534035584424894016340, −11.12286817891502239056199908432, −10.61764025978396345366747873232, −9.96158140475991653073232642359, −8.77288755648369126656006686476, −8.14710946478081833356607553517, −7.53933624830265341782340668038, −6.35123338312617427114257572507, −5.943018538827444712588245659099, −5.05619780844986831409832153316, −3.95372554687715689553224251249, −3.26996258155104796729745036029, −2.20227004697306607300394008649, −1.567064677564953401421710218598, −0.22642953882133355638913657558, 0.9601143476134383838007613562, 1.917432806151047850992516424111, 2.39683610843059746991961788555, 3.96650963408174734458543795658, 4.6848512809275902975703686646, 5.08808093814556282819400439312, 6.188504561162796839150845225874, 6.92525323275579586024188007032, 8.02004524681927869556918308355, 8.538632330816471987959288988693, 9.242294547117398895368105200555, 10.05190339889693272280295699814, 11.00140769570298532876295648120, 11.558240539450548140017482874332, 12.46618539968337036672930415890, 13.16547493504998288234141522209, 13.72246289924377640534250910628, 14.61618996403405058577139453371, 15.34398130002877185765179103428, 16.07324189153780009054881705625, 16.82357606009735160574928599251, 17.55362103662033820203297703288, 18.12068153723432060399529299668, 18.681325666881364274590118291571, 20.05768766186031600722177511874

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