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Properties

Label 1-5160-5160.227-r0-0-0
Degree $1$
Conductor $5160$
Sign $0.909 - 0.416i$
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 + (0.222 − 0.974i)11-s + (0.563 + 0.826i)13-s + (−0.997 − 0.0747i)17-s + (−0.733 + 0.680i)19-s + (−0.294 − 0.955i)23-s + (0.365 + 0.930i)29-s + (0.988 + 0.149i)31-s + (0.866 − 0.5i)37-s + (−0.623 + 0.781i)41-s + (0.974 − 0.222i)47-s + (0.5 + 0.866i)49-s + (0.563 − 0.826i)53-s + (−0.900 + 0.433i)59-s + (−0.988 + 0.149i)61-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.222 − 0.974i)11-s + (0.563 + 0.826i)13-s + (−0.997 − 0.0747i)17-s + (−0.733 + 0.680i)19-s + (−0.294 − 0.955i)23-s + (0.365 + 0.930i)29-s + (0.988 + 0.149i)31-s + (0.866 − 0.5i)37-s + (−0.623 + 0.781i)41-s + (0.974 − 0.222i)47-s + (0.5 + 0.866i)49-s + (0.563 − 0.826i)53-s + (−0.900 + 0.433i)59-s + (−0.988 + 0.149i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5160\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 43\)
Sign: $0.909 - 0.416i$
Analytic conductor: \(23.9629\)
Root analytic conductor: \(23.9629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5160} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5160,\ (0:\ ),\ 0.909 - 0.416i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.287903413 - 0.2809216132i\)
\(L(\frac12)\) \(\approx\) \(1.287903413 - 0.2809216132i\)
\(L(1)\) \(\approx\) \(0.9371412112 - 0.07302099627i\)
\(L(1)\) \(\approx\) \(0.9371412112 - 0.07302099627i\)

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 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (0.563 + 0.826i)T \)
17 \( 1 + (-0.997 - 0.0747i)T \)
19 \( 1 + (-0.733 + 0.680i)T \)
23 \( 1 + (-0.294 - 0.955i)T \)
29 \( 1 + (0.365 + 0.930i)T \)
31 \( 1 + (0.988 + 0.149i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (0.563 - 0.826i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (-0.988 + 0.149i)T \)
67 \( 1 + (-0.680 - 0.733i)T \)
71 \( 1 + (-0.955 - 0.294i)T \)
73 \( 1 + (0.563 + 0.826i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.930 + 0.365i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
97 \( 1 + (-0.974 - 0.222i)T \)
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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

−17.911557165091143134033839693533, −17.4630287173174417965336659595, −16.82349091110751465600142952578, −15.76268659496847961179342482448, −15.35978060181096959097108955996, −15.13227685754224115285058138963, −13.83471303856704705260252575415, −13.343289114287599675240707286969, −12.76621763542894143991796772920, −12.05123426566765000022882989883, −11.44226863211666549329642758404, −10.48629413808744006563820510842, −10.01649807612004403324033737074, −9.16880020699454047493275627961, −8.72599821091156713806904128456, −7.78539546981556915203868410447, −7.08748105445752174046362592187, −6.209679122001813349604485490687, −5.93217040292530205028914510670, −4.747475590603795669119276052380, −4.229706169437355579029373986523, −3.25855864858019871718007607169, −2.55103010361594524853368429097, −1.82987716946855408207274264934, −0.621121183841333546319284364189, 0.55023699593166541307721355988, 1.50378377420183760942376331944, 2.5171710220279130895534507281, 3.271261172076777113060135776193, 4.088648210402762399249388106165, 4.5329801872840719386093727566, 5.766587409841416037363476119846, 6.49411700387211370477686229978, 6.65748695672135331089759981342, 7.77572251846465314524112748025, 8.65857924145480887223514926464, 8.96492213795580509522481615767, 9.92819855253117470020181991899, 10.61501309013883673351762115732, 11.11054733258681941001496790244, 11.97113780952668651604926587803, 12.64351363112550496631270997422, 13.41867295027488123766680304106, 13.84612932251862575185931526152, 14.511855834130692581232261360933, 15.41562841836909563723464358486, 16.1016687551467420389973224816, 16.66213104247482229376451435240, 16.96934763798997841180505350644, 18.20562797727660781027924954291

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