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Properties

Label 1-1859-1859.32-r0-0-0
Degree $1$
Conductor $1859$
Sign $0.990 + 0.138i$
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  + (−0.979 − 0.200i)2-s + (0.987 − 0.160i)3-s + (0.919 + 0.391i)4-s + (−0.239 + 0.970i)5-s + (−0.999 − 0.0402i)6-s + (0.903 − 0.428i)7-s + (−0.822 − 0.568i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (0.970 + 0.239i)12-s + (−0.970 + 0.239i)14-s + (−0.0804 + 0.996i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (−0.992 + 0.120i)18-s + (−0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (−0.979 − 0.200i)2-s + (0.987 − 0.160i)3-s + (0.919 + 0.391i)4-s + (−0.239 + 0.970i)5-s + (−0.999 − 0.0402i)6-s + (0.903 − 0.428i)7-s + (−0.822 − 0.568i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (0.970 + 0.239i)12-s + (−0.970 + 0.239i)14-s + (−0.0804 + 0.996i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (−0.992 + 0.120i)18-s + (−0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.990 + 0.138i$
Analytic conductor: \(8.63315\)
Root analytic conductor: \(8.63315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1859,\ (0:\ ),\ 0.990 + 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.712353862 + 0.1195617416i\)
\(L(\frac12)\) \(\approx\) \(1.712353862 + 0.1195617416i\)
\(L(1)\) \(\approx\) \(1.110178238 + 0.007650108432i\)
\(L(1)\) \(\approx\) \(1.110178238 + 0.007650108432i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.979 - 0.200i)T \)
3 \( 1 + (0.987 - 0.160i)T \)
5 \( 1 + (-0.239 + 0.970i)T \)
7 \( 1 + (0.903 - 0.428i)T \)
17 \( 1 + (0.428 + 0.903i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.200 - 0.979i)T \)
31 \( 1 + (-0.464 - 0.885i)T \)
37 \( 1 + (0.534 - 0.845i)T \)
41 \( 1 + (0.160 + 0.987i)T \)
43 \( 1 + (-0.845 + 0.534i)T \)
47 \( 1 + (0.992 + 0.120i)T \)
53 \( 1 + (0.568 - 0.822i)T \)
59 \( 1 + (0.721 + 0.692i)T \)
61 \( 1 + (0.996 - 0.0804i)T \)
67 \( 1 + (0.391 + 0.919i)T \)
71 \( 1 + (-0.774 + 0.632i)T \)
73 \( 1 + (0.663 + 0.748i)T \)
79 \( 1 + (-0.120 + 0.992i)T \)
83 \( 1 + (0.935 - 0.354i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.960 - 0.278i)T \)
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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

−20.30716050439072009153974903576, −19.28405417697704902825974561779, −18.69660833359126396708656853149, −18.06317784825780648097804150822, −17.09214515849360519900341572154, −16.42797909889462909857401475314, −15.80877519774905106184331083124, −14.97884447962410239884349709156, −14.50847758984289938136322533990, −13.59406913135871571693929579214, −12.43659463086025699103333175753, −12.01913174700010916171483841872, −10.878706134487362758552414463255, −10.23745921884421738578107423978, −9.13709723455439547695221921955, −8.843979652899702866787938574828, −8.17330028854914803917657630275, −7.55775306907155043808441286527, −6.66029509925059894788176113561, −5.33611174613584138592995463408, −4.80031649420796661802721658822, −3.63808765524033915819492631690, −2.52935844784507723666990451588, −1.77308104733439891593980388237, −0.91846289434689922131013088752, 0.98735934323035632175398422551, 2.02157992030999394892277933122, 2.573846260201890686682060525992, 3.645871588404885211126104370740, 4.20105969349508262316244354413, 5.849082559372338217233869967415, 6.83864083386742845409749024603, 7.436767812650672901276481461220, 8.07055746154627138071947888462, 8.62901312995779374880810339345, 9.71187459592972524952071681954, 10.23505451847974579007282816254, 11.13569267040209152402972936024, 11.56575559176390285489143211338, 12.75745424033435192836772423459, 13.474995433621350152900983992873, 14.58510880940171527195071989843, 14.92536764144153637923067399755, 15.5373289858855398118955940618, 16.62414963515850565857633440574, 17.49885953091918200525067769089, 17.993555146166125960281157547528, 18.84764444379159909439441045757, 19.33165185840887510620906760304, 19.88334141406571776732997163455

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