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Properties

Label 2-855-95.69-c0-0-1
Degree $2$
Conductor $855$
Sign $-0.412 - 0.910i$
Analytic cond. $0.426700$
Root an. cond. $0.653223$
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  + (0.866 + 1.5i)2-s + (−1 + 1.73i)4-s + (0.866 − 0.5i)5-s − 1.73·8-s + (1.5 + 0.866i)10-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + 2i·20-s + (−1.73 − i)23-s + (0.499 − 0.866i)25-s − 1.73i·31-s + (−1.5 − 0.866i)34-s + (−0.866 + 1.5i)38-s + (−1.49 + 0.866i)40-s + ⋯
L(s)  = 1  + (0.866 + 1.5i)2-s + (−1 + 1.73i)4-s + (0.866 − 0.5i)5-s − 1.73·8-s + (1.5 + 0.866i)10-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + 2i·20-s + (−1.73 − i)23-s + (0.499 − 0.866i)25-s − 1.73i·31-s + (−1.5 − 0.866i)34-s + (−0.866 + 1.5i)38-s + (−1.49 + 0.866i)40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.412 - 0.910i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ -0.412 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.593655611\)
\(L(\frac12)\) \(\approx\) \(1.593655611\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - iT - T^{2} \)
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

−10.43480280904506147833154625622, −9.635180173636796167109736353895, −8.561723766346746106455900171501, −8.068913709889442605282781510102, −6.98779146233056330745673920035, −6.05950648777392315700101186591, −5.71704389578689437784486046228, −4.58409087335544454103413225110, −3.88009064443464377114960277673, −2.16515664564806636712381299776, 1.60445160836339725235751855805, 2.57566375606198824045203218610, 3.43934987110391055307515671405, 4.60178270106430705287959321432, 5.42240417030468864773521486574, 6.33562581187039753591729059942, 7.41789879881880681744226894164, 8.948706821942805750524587538093, 9.610955019864053806241615234638, 10.34567566725517144968049087535

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