Properties

Label 2-1859-143.120-c0-0-0
Degree $2$
Conductor $1859$
Sign $-0.990 + 0.134i$
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  + (0.900 + 1.56i)3-s + (−0.5 + 0.866i)4-s − 1.24·5-s + (−1.12 + 1.94i)9-s + (0.5 + 0.866i)11-s − 1.80·12-s + (−1.12 − 1.94i)15-s + (−0.499 − 0.866i)16-s + (0.623 − 1.07i)20-s + (0.222 + 0.385i)23-s + 0.554·25-s − 2.24·27-s + 0.445·31-s + (−0.900 + 1.56i)33-s + (−1.12 − 1.94i)36-s + (−0.900 − 1.56i)37-s + ⋯
L(s)  = 1  + (0.900 + 1.56i)3-s + (−0.5 + 0.866i)4-s − 1.24·5-s + (−1.12 + 1.94i)9-s + (0.5 + 0.866i)11-s − 1.80·12-s + (−1.12 − 1.94i)15-s + (−0.499 − 0.866i)16-s + (0.623 − 1.07i)20-s + (0.222 + 0.385i)23-s + 0.554·25-s − 2.24·27-s + 0.445·31-s + (−0.900 + 1.56i)33-s + (−1.12 − 1.94i)36-s + (−0.900 − 1.56i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-0.990 + 0.134i$
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: \((2,\ 1859,\ (\ :0),\ -0.990 + 0.134i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9216763726\)
\(L(\frac12)\) \(\approx\) \(0.9216763726\)
\(L(1)\) 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 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + 1.24T + T^{2} \)
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 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - 0.445T + T^{2} \)
37 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - 0.445T + T^{2} \)
53 \( 1 - 1.24T + T^{2} \)
59 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.727822088236610749029578723580, −8.778549665347243781739913178650, −8.658120345024665781057957104518, −7.61114929080680926971866250053, −7.17866930828601063426633275304, −5.38985382303255524494021876246, −4.47331552860339295384790494166, −4.01785958866756952172148102533, −3.48594262308510766057951267835, −2.49860207756640481287262127317, 0.63888356687725366145910625436, 1.68083895241522822737107859156, 3.01778991815372593514134567190, 3.79894262483043957234214970432, 4.91764685008333796867960319237, 6.19630690280850177477578155025, 6.63786991534231890075153757742, 7.60697896588974570630626085706, 8.243071878100391837260322544211, 8.744590157231603032968287111309

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