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Properties

Label 1-1859-1859.1005-r0-0-0
Degree $1$
Conductor $1859$
Sign $0.952 - 0.305i$
Analytic cond. $8.63315$
Root an. cond. $8.63315$
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  + (−0.247 + 0.968i)2-s + (−0.136 − 0.990i)3-s + (−0.877 − 0.478i)4-s + (−0.779 − 0.626i)5-s + (0.993 + 0.112i)6-s + (0.999 + 0.0161i)7-s + (0.681 − 0.732i)8-s + (−0.962 + 0.270i)9-s + (0.799 − 0.600i)10-s + (−0.354 + 0.935i)12-s + (−0.262 + 0.964i)14-s + (−0.513 + 0.857i)15-s + (0.541 + 0.840i)16-s + (0.818 − 0.574i)17-s + (−0.0241 − 0.999i)18-s + (−0.913 + 0.406i)19-s + ⋯
L(s)  = 1  + (−0.247 + 0.968i)2-s + (−0.136 − 0.990i)3-s + (−0.877 − 0.478i)4-s + (−0.779 − 0.626i)5-s + (0.993 + 0.112i)6-s + (0.999 + 0.0161i)7-s + (0.681 − 0.732i)8-s + (−0.962 + 0.270i)9-s + (0.799 − 0.600i)10-s + (−0.354 + 0.935i)12-s + (−0.262 + 0.964i)14-s + (−0.513 + 0.857i)15-s + (0.541 + 0.840i)16-s + (0.818 − 0.574i)17-s + (−0.0241 − 0.999i)18-s + (−0.913 + 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.952 - 0.305i$
Analytic conductor: \(8.63315\)
Root analytic conductor: \(8.63315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1859,\ (0:\ ),\ 0.952 - 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.098933200 - 0.1721812181i\)
\(L(\frac12)\) \(\approx\) \(1.098933200 - 0.1721812181i\)
\(L(1)\) \(\approx\) \(0.8284860341 + 0.008236743821i\)
\(L(1)\) \(\approx\) \(0.8284860341 + 0.008236743821i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.247 + 0.968i)T \)
3 \( 1 + (-0.136 - 0.990i)T \)
5 \( 1 + (-0.779 - 0.626i)T \)
7 \( 1 + (0.999 + 0.0161i)T \)
17 \( 1 + (0.818 - 0.574i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.997 - 0.0643i)T \)
31 \( 1 + (0.861 + 0.506i)T \)
37 \( 1 + (0.953 - 0.301i)T \)
41 \( 1 + (0.136 + 0.990i)T \)
43 \( 1 + (0.948 - 0.316i)T \)
47 \( 1 + (-0.0241 + 0.999i)T \)
53 \( 1 + (0.981 - 0.192i)T \)
59 \( 1 + (-0.966 - 0.254i)T \)
61 \( 1 + (-0.657 + 0.753i)T \)
67 \( 1 + (-0.428 + 0.903i)T \)
71 \( 1 + (0.827 + 0.561i)T \)
73 \( 1 + (0.989 - 0.144i)T \)
79 \( 1 + (-0.607 + 0.794i)T \)
83 \( 1 + (0.527 - 0.849i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.152 - 0.988i)T \)
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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

−20.06519726582473190558959679103, −19.63046847200236515463671074087, −18.80823566543873856887795667240, −18.00169005402842698140434234913, −17.30264005288781466688166107580, −16.70247011206766241031468934714, −15.59543519746734276366453530984, −15.00280606399289925418411211464, −14.2958938623848459296998846995, −13.61805413751809181994592975479, −12.20711942054320012052413882288, −11.880612753228784712511935382346, −10.97654436775341032333965406065, −10.65369716533286137640967316565, −9.892943393852528296284606163143, −8.94325819991592241138138083270, −8.15035372305580311391124508687, −7.68792008966161683693024878740, −6.25039250558617586922116553805, −5.1862254051932985858387370058, −4.36049841599371269909090528144, −3.88250473157283573010009784211, −2.97897412606313098875754798050, −2.12448368599205123444695806017, −0.76122551739168507201420686478, 0.72751805761211562287810066342, 1.37142738354074552955723298151, 2.674469567653305748806415784291, 4.16303500203095386412591770313, 4.75599494440718561078242297041, 5.62362289981401839045978728260, 6.410491426189862769085933991085, 7.321875379888183850843648094952, 8.03974498816992189517192584295, 8.28815114487975791322690019982, 9.13397942728233381587870317629, 10.34462803897022424904978157683, 11.2219711486570789330702594110, 12.15003311804374247127061577452, 12.57944442771218309977651152424, 13.590167538914714968313850209, 14.310613903458761579822169382056, 14.8123877564674603333996301620, 15.80355126508511130464242162493, 16.56022736931967081363450859684, 17.093568605735037983888684992292, 17.8936338498259033099035729688, 18.454105549228697119606553715, 19.18894592329528242735907082640, 19.80221536498931924903325925708

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