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Properties

Label 2-855-855.664-c0-0-5
Degree $2$
Conductor $855$
Sign $0.766 + 0.642i$
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  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.499 + 0.866i)6-s − 8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s − 0.999·15-s + (0.5 − 0.866i)16-s + 0.999·18-s + 19-s + (0.499 + 0.866i)22-s + (−0.499 + 0.866i)24-s + (−0.499 + 0.866i)25-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.499 + 0.866i)6-s − 8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s − 0.999·15-s + (0.5 − 0.866i)16-s + 0.999·18-s + 19-s + (0.499 + 0.866i)22-s + (−0.499 + 0.866i)24-s + (−0.499 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (664, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7872579036\)
\(L(\frac12)\) \(\approx\) \(0.7872579036\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 - T \)
good2 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.868351095495537992763089265994, −8.968412486621384362536473701498, −8.477507206788408641879294532296, −7.72873036252335400225208718436, −7.21425743079743222797291623599, −6.12109368317711078851245996097, −5.36499606257001995336925973340, −3.74856848021277261431932612987, −2.79417238625653623392222431050, −0.910952507530333710820793795314, 1.97736312379105568042916221952, 2.92812338595745336661144565351, 3.85018869333763373901578547749, 4.85841572712301605428615337350, 6.22175325819062784340835775617, 7.20878344629677365277336471919, 8.118696818140407307513335041412, 9.432149969288258726472015327291, 9.503887657488290941450810624542, 10.43110305780675066030312246768

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