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Properties

Label 2-855-95.9-c1-0-4
Degree $2$
Conductor $855$
Sign $-0.323 - 0.946i$
Analytic cond. $6.82720$
Root an. cond. $2.61289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 + 0.212i)2-s + (−0.477 + 0.173i)4-s + (−2.06 − 0.857i)5-s + (−3.28 − 1.89i)7-s + (2.65 − 1.53i)8-s + (2.66 + 0.593i)10-s + (−0.618 − 1.07i)11-s + (2.22 − 2.64i)13-s + (4.35 + 1.58i)14-s + (−2.08 + 1.75i)16-s + (2.96 − 0.522i)17-s + (−4.28 + 0.793i)19-s + (1.13 + 0.0503i)20-s + (0.971 + 1.15i)22-s + (−2.10 − 5.77i)23-s + ⋯
L(s)  = 1  + (−0.850 + 0.149i)2-s + (−0.238 + 0.0868i)4-s + (−0.923 − 0.383i)5-s + (−1.24 − 0.716i)7-s + (0.937 − 0.541i)8-s + (0.843 + 0.187i)10-s + (−0.186 − 0.323i)11-s + (0.616 − 0.734i)13-s + (1.16 + 0.423i)14-s + (−0.522 + 0.438i)16-s + (0.718 − 0.126i)17-s + (−0.983 + 0.181i)19-s + (0.253 + 0.0112i)20-s + (0.207 + 0.246i)22-s + (−0.438 − 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.323 - 0.946i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0873426 + 0.122110i\)
\(L(\frac12)\) \(\approx\) \(0.0873426 + 0.122110i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.06 + 0.857i)T \)
19 \( 1 + (4.28 - 0.793i)T \)
good2 \( 1 + (1.20 - 0.212i)T + (1.87 - 0.684i)T^{2} \)
7 \( 1 + (3.28 + 1.89i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.618 + 1.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.22 + 2.64i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.96 + 0.522i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (2.10 + 5.77i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.744 - 4.22i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.55 - 4.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.13iT - 37T^{2} \)
41 \( 1 + (4.08 - 3.42i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.12 + 8.57i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (7.19 + 1.26i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-1.13 - 3.13i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.141 + 0.804i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (6.01 - 2.18i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.995 + 0.175i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-12.8 - 4.67i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-7.11 - 8.47i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-1.06 + 0.889i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.18 - 1.26i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.06 - 1.73i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (3.01 - 0.531i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.40412576481394211312888769554, −9.552818589715252791366654518144, −8.502759113029102901374778757062, −8.213170911253176768773113788272, −7.17133647983675797924758543545, −6.44140305042166537449000546926, −5.02822699864876667829394176496, −3.91306516425723846446727400917, −3.27654046501880728168194667762, −0.957254292837443668752785662365, 0.13029670865480457067332755841, 2.04739189453872193420387072302, 3.44752272228814154831538771668, 4.30812160196174411373739879009, 5.67542798575370156628649302024, 6.58910342229399716742787491661, 7.62529530291001269929516891099, 8.285293958721414963019887995353, 9.316157077962388438274617334397, 9.626561722420790879054573341933

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