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Properties

Label 2-1859-143.120-c0-0-4
Degree $2$
Conductor $1859$
Sign $0.611 + 0.791i$
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  + (0.222 + 0.385i)3-s + (−0.5 + 0.866i)4-s − 1.80·5-s + (0.400 − 0.694i)9-s + (−0.5 − 0.866i)11-s − 0.445·12-s + (−0.400 − 0.694i)15-s + (−0.499 − 0.866i)16-s + (0.900 − 1.56i)20-s + (−0.623 − 1.07i)23-s + 2.24·25-s + 0.801·27-s + 1.24·31-s + (0.222 − 0.385i)33-s + (0.400 + 0.694i)36-s + (0.222 + 0.385i)37-s + ⋯
L(s)  = 1  + (0.222 + 0.385i)3-s + (−0.5 + 0.866i)4-s − 1.80·5-s + (0.400 − 0.694i)9-s + (−0.5 − 0.866i)11-s − 0.445·12-s + (−0.400 − 0.694i)15-s + (−0.499 − 0.866i)16-s + (0.900 − 1.56i)20-s + (−0.623 − 1.07i)23-s + 2.24·25-s + 0.801·27-s + 1.24·31-s + (0.222 − 0.385i)33-s + (0.400 + 0.694i)36-s + (0.222 + 0.385i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.611 + 0.791i$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1836, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ 0.611 + 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5545290149\)
\(L(\frac12)\) \(\approx\) \(0.5545290149\)
\(L(1)\) 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 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + 1.80T + T^{2} \)
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 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - 1.24T + T^{2} \)
37 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - 1.24T + T^{2} \)
53 \( 1 + 1.80T + T^{2} \)
59 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.052528285256124664942210525774, −8.343755916964668530521849920720, −8.031949636629773420434394013196, −7.18292607131646332484832153029, −6.29538516758907923194441289523, −4.75104560669187817785022607203, −4.29894685086623867264432826421, −3.46096774660225036332750638928, −2.94770158982838451215935443044, −0.45652607919955912686014237564, 1.29158706325489900435117947455, 2.63194283799297154669178767017, 3.98445062448233371355586929358, 4.53653097180458638557719166258, 5.30693685541164579824621412461, 6.52815579065689873358125245535, 7.47699775098025987586347486052, 7.81823045500978668737531690053, 8.592916730391776849903505061759, 9.557642979360547403480360406337

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