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Properties

Label 2-1859-143.87-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8851024649\)
\(L(\frac12)\) \(\approx\) \(0.8851024649\)
\(L(1)\) 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

−9.671465996428387862732459896547, −8.837701924514054797278513765740, −7.75102925300727901955427970732, −7.50281322143168091110421891058, −6.04982071166272177233153696979, −5.20972071855151216091272961510, −4.53828644821471292833844211914, −3.90658335861686158128503077025, −2.00717409566619480561942720865, −1.30231167670623202038777097327, 1.03793091975720705567857414153, 2.24547349066207240307558884221, 3.73315915290590532276819852297, 4.42260721074976835452459495415, 5.63800918306564341762373137537, 6.32468435938000764370332451689, 7.24047690053527872539143945547, 7.946268550187488961865676673091, 8.549611134995551109059527612398, 9.176853829501013523931520062408

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