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Properties

Label 2-1859-143.87-c0-0-1
Degree $2$
Conductor $1859$
Sign $-0.999 + 0.0128i$
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  + (0.535 + 0.309i)2-s + (−0.309 + 0.535i)3-s + (−0.309 − 0.535i)4-s + (−0.330 + 0.190i)6-s + (−1.40 + 0.809i)7-s − 0.999i·8-s + (0.309 + 0.535i)9-s + (−0.866 − 0.5i)11-s + 0.381·12-s − 14-s + 0.381i·18-s + (−0.535 + 0.309i)19-s − i·21-s + (−0.309 − 0.535i)22-s + (−0.809 + 1.40i)23-s + (0.535 + 0.309i)24-s + ⋯
L(s)  = 1  + (0.535 + 0.309i)2-s + (−0.309 + 0.535i)3-s + (−0.309 − 0.535i)4-s + (−0.330 + 0.190i)6-s + (−1.40 + 0.809i)7-s − 0.999i·8-s + (0.309 + 0.535i)9-s + (−0.866 − 0.5i)11-s + 0.381·12-s − 14-s + 0.381i·18-s + (−0.535 + 0.309i)19-s − i·21-s + (−0.309 − 0.535i)22-s + (−0.809 + 1.40i)23-s + (0.535 + 0.309i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-0.999 + 0.0128i$
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: \((2,\ 1859,\ (\ :0),\ -0.999 + 0.0128i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3079718098\)
\(L(\frac12)\) \(\approx\) \(0.3079718098\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 \)
good2 \( 1 + (-0.535 - 0.309i)T + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (1.40 - 0.809i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.535 - 0.309i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.40 + 0.809i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - 0.618iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.61iT - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.973000888640401572025764327140, −9.261493749808808747137646313007, −8.312961777954797807031946997623, −7.29092393996187886894299321804, −6.30943895161998876408386061435, −5.63544413042699576444749361359, −5.27746476632407186688189148697, −4.08698321424054524812220939389, −3.38725381702254621835130983226, −2.05986217996456726110407011412, 0.18385131770699722780434294840, 2.15237883155234490761542186277, 3.19745163851123231445037232704, 3.98096519049005469572333588841, 4.72112316929006317912341035959, 5.94965653196301273070783344782, 6.62831538189258980785098328560, 7.36046988571913609946652446805, 8.109289376955329770952741714506, 9.078179341100794498149264832471

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