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Properties

Label 2-1859-143.120-c0-0-8
Degree $2$
Conductor $1859$
Sign $-0.978 - 0.207i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-0.978 - 0.207i$
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: \((2,\ 1859,\ (\ :0),\ -0.978 - 0.207i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1249251677\)
\(L(\frac12)\) \(\approx\) \(0.1249251677\)
\(L(1)\) 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

−8.901547589637061661996194598758, −7.926799493926351501926670007263, −7.54420638872519622815817375703, −6.84431595719118666169371843956, −5.76949214350841778841558898958, −5.13062689841110978456685497293, −3.81650923369002207303425411643, −3.18705813254827142067393018524, −1.70675833011910832240145033310, −0.099316330147147217617698363089, 1.82559529387583970240012192689, 3.37906710768151041114299336926, 4.55701244258842335996175365646, 4.73549139601209432308764670041, 5.61230994939004764608229918792, 6.49181269153000681068934040356, 7.47955205958379332604149458569, 8.473310066454850743355843026205, 9.339713896257786726835122240232, 9.893322350512661350337113362754

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