↑

Properties

Label 2-1859-143.10-c0-0-7
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.623 − 1.07i)3-s + (0.5 − 0.866i)4-s − 0.445i·5-s + (−0.277 + 0.480i)9-s + (−0.866 + 0.5i)11-s − 1.24·12-s + (−0.480 + 0.277i)15-s + (−0.499 − 0.866i)16-s + (−0.385 − 0.222i)20-s + (−0.900 − 1.56i)23-s + 0.801·25-s − 0.554·27-s − 1.80i·31-s + (1.07 + 0.623i)33-s + (0.277 + 0.480i)36-s + (−1.07 + 0.623i)37-s + ⋯
L(s)  = 1  + (−0.623 − 1.07i)3-s + (0.5 − 0.866i)4-s − 0.445i·5-s + (−0.277 + 0.480i)9-s + (−0.866 + 0.5i)11-s − 1.24·12-s + (−0.480 + 0.277i)15-s + (−0.499 − 0.866i)16-s + (−0.385 − 0.222i)20-s + (−0.900 − 1.56i)23-s + 0.801·25-s − 0.554·27-s − 1.80i·31-s + (1.07 + 0.623i)33-s + (0.277 + 0.480i)36-s + (−1.07 + 0.623i)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: $\chi_{1859} (868, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ -0.978 + 0.207i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8245403629\)
\(L(\frac12)\) \(\approx\) \(0.8245403629\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + 0.445iT - T^{2} \)
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 \( 1 + 1.80iT - T^{2} \)
37 \( 1 + (1.07 - 0.623i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - 1.80iT - T^{2} \)
53 \( 1 + 0.445T + T^{2} \)
59 \( 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (1.07 + 0.623i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.56 + 0.900i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.385 - 0.222i)T + (0.5 + 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.104609676391076450019523731337, −8.018041418758205411770243307080, −7.41115405926245216489799511737, −6.51876232881087138887419509197, −6.05901827675359064611378448646, −5.19237373555233662611775779442, −4.39148343576086726193323055563, −2.62976208698556994201985775085, −1.80703256421689169991640182158, −0.62859277617914111008779719050, 2.12735285947976037677354338385, 3.34537235158205384025104382988, 3.78403524938029026804899530963, 5.05652149349219436260839565310, 5.56954082605486785455332844540, 6.71144075543178942687685228582, 7.36516197040352917142177012597, 8.275794626746154236260520805426, 8.968473667551713724219344276427, 10.18074552214496218214307839235

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