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Properties

Label 1-5160-5160.587-r0-0-0
Degree $1$
Conductor $5160$
Sign $0.860 - 0.510i$
Analytic cond. $23.9629$
Root an. cond. $23.9629$
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  + (0.866 − 0.5i)7-s + (0.900 + 0.433i)11-s + (0.930 + 0.365i)13-s + (−0.149 + 0.988i)17-s + (0.0747 − 0.997i)19-s + (−0.563 − 0.826i)23-s + (−0.733 + 0.680i)29-s + (−0.955 − 0.294i)31-s + (−0.866 − 0.5i)37-s + (0.222 + 0.974i)41-s + (−0.433 − 0.900i)47-s + (0.5 − 0.866i)49-s + (0.930 − 0.365i)53-s + (0.623 − 0.781i)59-s + (0.955 − 0.294i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)7-s + (0.900 + 0.433i)11-s + (0.930 + 0.365i)13-s + (−0.149 + 0.988i)17-s + (0.0747 − 0.997i)19-s + (−0.563 − 0.826i)23-s + (−0.733 + 0.680i)29-s + (−0.955 − 0.294i)31-s + (−0.866 − 0.5i)37-s + (0.222 + 0.974i)41-s + (−0.433 − 0.900i)47-s + (0.5 − 0.866i)49-s + (0.930 − 0.365i)53-s + (0.623 − 0.781i)59-s + (0.955 − 0.294i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5160\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 43\)
Sign: $0.860 - 0.510i$
Analytic conductor: \(23.9629\)
Root analytic conductor: \(23.9629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5160} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5160,\ (0:\ ),\ 0.860 - 0.510i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.144950835 - 0.5882907998i\)
\(L(\frac12)\) \(\approx\) \(2.144950835 - 0.5882907998i\)
\(L(1)\) \(\approx\) \(1.285359759 - 0.09664995682i\)
\(L(1)\) \(\approx\) \(1.285359759 - 0.09664995682i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
43 \( 1 \)
good7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (0.930 + 0.365i)T \)
17 \( 1 + (-0.149 + 0.988i)T \)
19 \( 1 + (0.0747 - 0.997i)T \)
23 \( 1 + (-0.563 - 0.826i)T \)
29 \( 1 + (-0.733 + 0.680i)T \)
31 \( 1 + (-0.955 - 0.294i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.433 - 0.900i)T \)
53 \( 1 + (0.930 - 0.365i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (0.955 - 0.294i)T \)
67 \( 1 + (0.997 + 0.0747i)T \)
71 \( 1 + (-0.826 - 0.563i)T \)
73 \( 1 + (0.930 + 0.365i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.680 - 0.733i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
97 \( 1 + (0.433 - 0.900i)T \)
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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

−18.10126494294793351943659553779, −17.4524186817249428127521664246, −16.74667011812259843045258861644, −15.97992482710108120528698788444, −15.48773488320977403682192104240, −14.61660136366494102630101139537, −14.11294835583768536061674124021, −13.54367959672095910035087215688, −12.66617906095244084101079987379, −11.77522491719869430567920562856, −11.5385734283230670783894599856, −10.779254020767323611943249359176, −9.94361072842052551215547191439, −9.09687018319196543125714433771, −8.651089038749339553139327013081, −7.86373448352611063357197298281, −7.24038381788603021908116371571, −6.2390071049272906746801768213, −5.640657481910324965196134728367, −5.07969384122217120050461194067, −3.90377388833740541183339383852, −3.61409417510125538794665945939, −2.44005949554121211955378709214, −1.655150002242351100297333801723, −0.93870371223926897241458090475, 0.68391102291633246127449490748, 1.71318527644370484750245296373, 2.06218129851828956542672748702, 3.52889387618003011864341863768, 3.951841258935923971284755008117, 4.69800820239418354752484908664, 5.47844046587550730507753324802, 6.42879782063689100391480122109, 6.92165586848371305665023405339, 7.71621464593834698043104722937, 8.59927860909919793849767801823, 8.92092014281057133615519104145, 9.905978856015547155245809942674, 10.65845789831859512396408234423, 11.25526735077347247497199308224, 11.72848656948075666183257967165, 12.71439805798616461433347959330, 13.23392907832358900033341291231, 14.08217906505627819306506817180, 14.61780262635070262702231025077, 15.09758853120737819725786670543, 16.05926405971020971220258771512, 16.66768343977550366913914181180, 17.300117444060527913751227131286, 17.91815723999151585507927544519

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