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Properties

Label 1-1859-1859.537-r0-0-0
Degree $1$
Conductor $1859$
Sign $0.924 + 0.381i$
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  + (−0.966 + 0.254i)2-s + (−0.384 + 0.923i)3-s + (0.870 − 0.493i)4-s + (0.681 + 0.732i)5-s + (0.136 − 0.990i)6-s + (0.769 + 0.638i)7-s + (−0.715 + 0.698i)8-s + (−0.704 − 0.709i)9-s + (−0.845 − 0.534i)10-s + (0.120 + 0.992i)12-s + (−0.906 − 0.421i)14-s + (−0.937 + 0.347i)15-s + (0.513 − 0.857i)16-s + (0.369 − 0.929i)17-s + (0.861 + 0.506i)18-s + (−0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (−0.966 + 0.254i)2-s + (−0.384 + 0.923i)3-s + (0.870 − 0.493i)4-s + (0.681 + 0.732i)5-s + (0.136 − 0.990i)6-s + (0.769 + 0.638i)7-s + (−0.715 + 0.698i)8-s + (−0.704 − 0.709i)9-s + (−0.845 − 0.534i)10-s + (0.120 + 0.992i)12-s + (−0.906 − 0.421i)14-s + (−0.937 + 0.347i)15-s + (0.513 − 0.857i)16-s + (0.369 − 0.929i)17-s + (0.861 + 0.506i)18-s + (−0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.924 + 0.381i$
Analytic conductor: \(8.63315\)
Root analytic conductor: \(8.63315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1859,\ (0:\ ),\ 0.924 + 0.381i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9195400428 + 0.1820486567i\)
\(L(\frac12)\) \(\approx\) \(0.9195400428 + 0.1820486567i\)
\(L(1)\) \(\approx\) \(0.6818398756 + 0.2660459381i\)
\(L(1)\) \(\approx\) \(0.6818398756 + 0.2660459381i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.966 + 0.254i)T \)
3 \( 1 + (-0.384 + 0.923i)T \)
5 \( 1 + (0.681 + 0.732i)T \)
7 \( 1 + (0.769 + 0.638i)T \)
17 \( 1 + (0.369 - 0.929i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.932 - 0.362i)T \)
31 \( 1 + (-0.644 - 0.764i)T \)
37 \( 1 + (0.827 - 0.561i)T \)
41 \( 1 + (0.384 - 0.923i)T \)
43 \( 1 + (0.278 - 0.960i)T \)
47 \( 1 + (0.861 - 0.506i)T \)
53 \( 1 + (-0.443 - 0.896i)T \)
59 \( 1 + (-0.0884 + 0.996i)T \)
61 \( 1 + (-0.554 - 0.832i)T \)
67 \( 1 + (0.200 - 0.979i)T \)
71 \( 1 + (0.877 + 0.478i)T \)
73 \( 1 + (0.998 + 0.0483i)T \)
79 \( 1 + (0.215 + 0.976i)T \)
83 \( 1 + (0.943 + 0.331i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.293 + 0.955i)T \)
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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

−19.986063609660672145648707710659, −19.32019643090641607999981043736, −18.40767879414422717158163484653, −17.89487352183648593404125946842, −17.159046112555518408802313396320, −16.85757702839960030085025086655, −16.14012392641985339738954238855, −14.82610145475959835300942954933, −14.09941220807184836242361054905, −13.08667995534139776856149965561, −12.6416542792208544583293501314, −11.839008319260107193670860064304, −11.00885181801420702726000500295, −10.43813004526773140659654358559, −9.52129498102656466176560510499, −8.624053334710276532123995421737, −7.910429363599876570494744340229, −7.481965221160477226438251253472, −6.28077923238175584650147913055, −5.83567850360778527303893494847, −4.70520818974480737865085813801, −3.557207682174728184715801705642, −2.20703477366307772332794102227, −1.54960691764290060570180656717, −1.046925933165707148626568123562, 0.53185026877434190082320989281, 2.14142621577102935830921486893, 2.51746397426178919505668758539, 3.80614430479029956167577223277, 5.05598343440741775148527058861, 5.65348695949237513733065180756, 6.35292574527897633914202989651, 7.261867158613514622059810237952, 8.18270979241304267688927489527, 9.243790975508461000828015813644, 9.39669698871960487370825386523, 10.490017110868307915276295805640, 10.96312446629513964843424196934, 11.55005874164234420409703969418, 12.44087754271292757495734411481, 13.91808727300740532341054803399, 14.563657386926111745461485039281, 15.1736098452977684527930979090, 15.71276278150742179953555896111, 16.77595613751477756516507058533, 17.12993673969230433782892854317, 18.156927149644643326360420953, 18.27289255342758170463854423313, 19.22513392820030306098014971552, 20.36344976837394823833972955655

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