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Properties

Label 2-855-19.11-c1-0-3
Degree $2$
Conductor $855$
Sign $-0.612 - 0.790i$
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  + (−0.145 + 0.251i)2-s + (0.957 + 1.65i)4-s + (0.5 − 0.866i)5-s − 0.486·7-s − 1.13·8-s + (0.145 + 0.251i)10-s − 5.34·11-s + (1.24 + 2.15i)13-s + (0.0707 − 0.122i)14-s + (−1.75 + 3.03i)16-s + (−1.70 + 2.95i)17-s + (−2.46 + 3.59i)19-s + 1.91·20-s + (0.777 − 1.34i)22-s + (3.20 + 5.55i)23-s + ⋯
L(s)  = 1  + (−0.102 + 0.178i)2-s + (0.478 + 0.829i)4-s + (0.223 − 0.387i)5-s − 0.183·7-s − 0.402·8-s + (0.0459 + 0.0796i)10-s − 1.61·11-s + (0.344 + 0.597i)13-s + (0.0189 − 0.0327i)14-s + (−0.437 + 0.757i)16-s + (−0.414 + 0.717i)17-s + (−0.565 + 0.825i)19-s + 0.428·20-s + (0.165 − 0.287i)22-s + (0.668 + 1.15i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.612 - 0.790i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.612 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.492122 + 1.00435i\)
\(L(\frac12)\) \(\approx\) \(0.492122 + 1.00435i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (2.46 - 3.59i)T \)
good2 \( 1 + (0.145 - 0.251i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + 0.486T + 7T^{2} \)
11 \( 1 + 5.34T + 11T^{2} \)
13 \( 1 + (-1.24 - 2.15i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.70 - 2.95i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.20 - 5.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.38 - 2.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.83T + 31T^{2} \)
37 \( 1 + 6.31T + 37T^{2} \)
41 \( 1 + (-1.29 + 2.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.24 + 2.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.55 - 9.62i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.49 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.36 + 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.92 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.58 - 13.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.96 - 8.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.50 + 9.53i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.06 - 7.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.86T + 83T^{2} \)
89 \( 1 + (3.25 + 5.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.58539434245057771840490817271, −9.559002939347053813812887450704, −8.547969898370372098738857458045, −8.056501244426341446235547071878, −7.12431786674917558533953253646, −6.22518783245651658566675423442, −5.27552217310230911694476247328, −4.08437095595011331828871110763, −3.01593181329600836600305933497, −1.88217393351276551773856321883, 0.51779915152568459489364373067, 2.36363389935450300142264967125, 2.91451809567342662777303383941, 4.72053066229396410016320458901, 5.47675882515386488474807177059, 6.44799121110183099749041722632, 7.13554180677777775331383644364, 8.241902538173743186722697755733, 9.154288669004501397091602183864, 10.24526682669309230737741205681

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