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Properties

Label 2-1859-143.120-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  + (−1.40 + 0.809i)2-s + (0.809 + 1.40i)3-s + (0.809 − 1.40i)4-s + (−2.26 − 1.30i)6-s + (0.535 + 0.309i)7-s + 0.999i·8-s + (−0.809 + 1.40i)9-s + (−0.866 + 0.5i)11-s + 2.61·12-s − 14-s − 2.61i·18-s + (1.40 + 0.809i)19-s + i·21-s + (0.809 − 1.40i)22-s + (0.309 + 0.535i)23-s + (−1.40 + 0.809i)24-s + ⋯
L(s)  = 1  + (−1.40 + 0.809i)2-s + (0.809 + 1.40i)3-s + (0.809 − 1.40i)4-s + (−2.26 − 1.30i)6-s + (0.535 + 0.309i)7-s + 0.999i·8-s + (−0.809 + 1.40i)9-s + (−0.866 + 0.5i)11-s + 2.61·12-s − 14-s − 2.61i·18-s + (1.40 + 0.809i)19-s + i·21-s + (0.809 − 1.40i)22-s + (0.309 + 0.535i)23-s + (−1.40 + 0.809i)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: $\chi_{1859} (1836, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ -0.999 - 0.0128i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7157948420\)
\(L(\frac12)\) \(\approx\) \(0.7157948420\)
\(L(1)\) 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 \( 1 + (1.40 - 0.809i)T + (0.5 - 0.866i)T^{2} \)
3 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (-0.535 - 0.309i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.40 - 0.809i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.535 + 0.309i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.61T + T^{2} \)
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 \( 1 - 1.61iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 0.618iT - T^{2} \)
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

−9.670772884479676274130578157881, −9.143265845437086993667817083983, −8.162105771322020533444942419063, −7.942136744818571350303656732416, −7.08259064169642642732652845060, −5.72438486148850312051429941237, −5.16387753540631044962001302349, −4.06394583016603531838096789264, −3.02089699436014202421841780159, −1.75527666158578997095656523757, 0.78703743845062292261249080368, 1.73737030934259849813176232717, 2.64829244729479229656926841380, 3.28752648419191950848433095489, 4.96142588500722692909356336985, 6.20417397094362371751105098414, 7.31323270957533547841230772617, 7.71334890094685066320185345086, 8.191455129068152655232679072947, 8.986461538512646359125767717879

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