Properties

Label 2-855-19.4-c1-0-0
Degree $2$
Conductor $855$
Sign $-0.939 - 0.343i$
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  + (−1.89 − 0.691i)2-s + (1.59 + 1.34i)4-s + (0.766 − 0.642i)5-s + (−1.54 + 2.67i)7-s + (−0.0878 − 0.152i)8-s + (−1.89 + 0.691i)10-s + (0.481 + 0.834i)11-s + (−0.513 + 2.91i)13-s + (4.78 − 4.01i)14-s + (−0.663 − 3.76i)16-s + (0.0366 + 0.0133i)17-s + (−4.31 − 0.596i)19-s + 2.08·20-s + (−0.338 − 1.91i)22-s + (−2.97 − 2.49i)23-s + ⋯
L(s)  = 1  + (−1.34 − 0.488i)2-s + (0.799 + 0.670i)4-s + (0.342 − 0.287i)5-s + (−0.584 + 1.01i)7-s + (−0.0310 − 0.0538i)8-s + (−0.600 + 0.218i)10-s + (0.145 + 0.251i)11-s + (−0.142 + 0.808i)13-s + (1.27 − 1.07i)14-s + (−0.165 − 0.940i)16-s + (0.00887 + 0.00323i)17-s + (−0.990 − 0.136i)19-s + 0.466·20-s + (−0.0720 − 0.408i)22-s + (−0.620 − 0.520i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.939 - 0.343i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.939 - 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00735523 + 0.0415021i\)
\(L(\frac12)\) \(\approx\) \(0.00735523 + 0.0415021i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.31 + 0.596i)T \)
good2 \( 1 + (1.89 + 0.691i)T + (1.53 + 1.28i)T^{2} \)
7 \( 1 + (1.54 - 2.67i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.481 - 0.834i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.513 - 2.91i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.0366 - 0.0133i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (2.97 + 2.49i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (8.76 - 3.19i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-4.68 + 8.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.11T + 37T^{2} \)
41 \( 1 + (2.09 + 11.8i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (3.79 - 3.18i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-7.53 + 2.74i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (4.64 + 3.89i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-2.60 - 0.948i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (7.68 + 6.44i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (8.78 - 3.19i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (9.64 - 8.09i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.155 - 0.883i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.54 - 8.75i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (4.80 - 8.31i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.51 + 8.61i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (7.31 + 2.66i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.29533004682933068117033167196, −9.563511921213207852002139895806, −9.016529922683902533227182672115, −8.448893681670000334283581731136, −7.36464457544442442375591048909, −6.36805345279583360945628695797, −5.42311025265904269161438782697, −4.13780953447665482469329621810, −2.52553177419025236291947497826, −1.82836967619592702105782387186, 0.03156016782010521147885970706, 1.50314339569445336468794478788, 3.18574364272067173992496979696, 4.30189278066795494703747552421, 5.86590427599427967893675254334, 6.58502077822751553185695614091, 7.41145093187799625077850207494, 8.054440694607076805478517790726, 9.001886895209614848091693629254, 9.795065011893696194725037694480

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