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Properties

Label 2-1859-143.87-c0-0-4
Degree $2$
Conductor $1859$
Sign $0.997 + 0.0743i$
Analytic cond. $0.927761$
Root an. cond. $0.963203$
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  + (−0.623 + 1.07i)3-s + (−0.5 − 0.866i)4-s + 0.445·5-s + (−0.277 − 0.480i)9-s + (0.5 − 0.866i)11-s + 1.24·12-s + (−0.277 + 0.480i)15-s + (−0.499 + 0.866i)16-s + (−0.222 − 0.385i)20-s + (0.900 − 1.56i)23-s − 0.801·25-s − 0.554·27-s + 1.80·31-s + (0.623 + 1.07i)33-s + (−0.277 + 0.480i)36-s + (0.623 − 1.07i)37-s + ⋯
L(s)  = 1  + (−0.623 + 1.07i)3-s + (−0.5 − 0.866i)4-s + 0.445·5-s + (−0.277 − 0.480i)9-s + (0.5 − 0.866i)11-s + 1.24·12-s + (−0.277 + 0.480i)15-s + (−0.499 + 0.866i)16-s + (−0.222 − 0.385i)20-s + (0.900 − 1.56i)23-s − 0.801·25-s − 0.554·27-s + 1.80·31-s + (0.623 + 1.07i)33-s + (−0.277 + 0.480i)36-s + (0.623 − 1.07i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.997 + 0.0743i$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ 0.997 + 0.0743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9063723651\)
\(L(\frac12)\) \(\approx\) \(0.9063723651\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - 0.445T + T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - 1.80T + T^{2} \)
37 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - 1.80T + T^{2} \)
53 \( 1 + 0.445T + T^{2} \)
59 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.508419019715048210265770449417, −8.973014509282867355699069972971, −8.097823038966850181505804740065, −6.66084026066184654081025110937, −6.00187454203816990084579354249, −5.39521957652060785700901466208, −4.54513207177709199599726641901, −3.96021448402681239518612304736, −2.51502387273918812684842421364, −0.910403748579808583283472202124, 1.20862475677991773526532489167, 2.36536180799443466949155882835, 3.58248623062528508787844967306, 4.58326091166527361973185677157, 5.49303302276095066179593624009, 6.43698357121179103462054350193, 7.09237698044942007025878474989, 7.70830576893491226952158419947, 8.513338665063175819115351140374, 9.534889709872140009227953596808

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