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Properties

Label 2-855-19.5-c1-0-16
Degree $2$
Conductor $855$
Sign $0.933 - 0.359i$
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  + (1.58 − 0.575i)2-s + (0.639 − 0.536i)4-s + (0.766 + 0.642i)5-s + (0.274 + 0.474i)7-s + (−0.981 + 1.69i)8-s + (1.58 + 0.575i)10-s + (0.165 − 0.286i)11-s + (0.837 + 4.74i)13-s + (0.707 + 0.593i)14-s + (−0.863 + 4.89i)16-s + (4.96 − 1.80i)17-s + (4.30 − 0.670i)19-s + 0.834·20-s + (0.0968 − 0.549i)22-s + (0.850 − 0.713i)23-s + ⋯
L(s)  = 1  + (1.11 − 0.407i)2-s + (0.319 − 0.268i)4-s + (0.342 + 0.287i)5-s + (0.103 + 0.179i)7-s + (−0.346 + 0.600i)8-s + (0.500 + 0.182i)10-s + (0.0499 − 0.0864i)11-s + (0.232 + 1.31i)13-s + (0.189 + 0.158i)14-s + (−0.215 + 1.22i)16-s + (1.20 − 0.438i)17-s + (0.988 − 0.153i)19-s + 0.186·20-s + (0.0206 − 0.117i)22-s + (0.177 − 0.148i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.933 - 0.359i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.933 - 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.79317 + 0.519710i\)
\(L(\frac12)\) \(\approx\) \(2.79317 + 0.519710i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (-4.30 + 0.670i)T \)
good2 \( 1 + (-1.58 + 0.575i)T + (1.53 - 1.28i)T^{2} \)
7 \( 1 + (-0.274 - 0.474i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.165 + 0.286i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.837 - 4.74i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-4.96 + 1.80i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-0.850 + 0.713i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (3.01 + 1.09i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.01 - 5.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.67T + 37T^{2} \)
41 \( 1 + (-1.37 + 7.79i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.25 - 1.05i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (4.32 + 1.57i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (5.15 - 4.32i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (6.39 - 2.32i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.520 + 0.436i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-7.30 - 2.65i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-0.832 - 0.698i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-2.42 + 13.7i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.243 - 1.38i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.427 + 0.740i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.52 + 14.3i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-6.59 + 2.40i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.40761377366092736229591037983, −9.395893034089577983212972900007, −8.700424214057694478915540706885, −7.51386370511313934791665769697, −6.55450491125411308799596081889, −5.56835728821320821067359938544, −4.89872598172084773399793685759, −3.76946826715701928158501240172, −2.96127596027046497058845427676, −1.73471076375379939021056895179, 1.10047973249215099581522493131, 3.01022401831637769703626618868, 3.81115053822249594175985995512, 5.02696842916046767046100237874, 5.56454089562266103700629352345, 6.33133002062596076816184415455, 7.49306687836902865174267324940, 8.188891653053125959634014213787, 9.506843609918766976858116716008, 9.985672458863693385763970163815

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