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Properties

Label 2-855-19.6-c1-0-22
Degree $2$
Conductor $855$
Sign $0.817 + 0.575i$
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 − 1.00i)2-s + (0.0809 − 0.459i)4-s + (0.173 + 0.984i)5-s + (2.00 − 3.47i)7-s + (1.20 + 2.08i)8-s + (1.20 + 1.00i)10-s + (1.38 + 2.39i)11-s + (−2.61 + 0.953i)13-s + (−1.09 − 6.20i)14-s + (4.43 + 1.61i)16-s + (2.76 − 2.31i)17-s + (1.79 − 3.97i)19-s + 0.466·20-s + (4.08 + 1.48i)22-s + (−0.237 + 1.34i)23-s + ⋯
L(s)  = 1  + (0.850 − 0.713i)2-s + (0.0404 − 0.229i)4-s + (0.0776 + 0.440i)5-s + (0.758 − 1.31i)7-s + (0.425 + 0.737i)8-s + (0.380 + 0.319i)10-s + (0.417 + 0.722i)11-s + (−0.726 + 0.264i)13-s + (−0.292 − 1.65i)14-s + (1.10 + 0.403i)16-s + (0.670 − 0.562i)17-s + (0.410 − 0.911i)19-s + 0.104·20-s + (0.870 + 0.316i)22-s + (−0.0495 + 0.280i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.817 + 0.575i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (766, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.817 + 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63508 - 0.834049i\)
\(L(\frac12)\) \(\approx\) \(2.63508 - 0.834049i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-1.79 + 3.97i)T \)
good2 \( 1 + (-1.20 + 1.00i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (-2.00 + 3.47i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.38 - 2.39i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.61 - 0.953i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.76 + 2.31i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.237 - 1.34i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-7.28 - 6.11i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.776 - 1.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.51T + 37T^{2} \)
41 \( 1 + (6.21 + 2.26i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.08 + 6.15i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (6.97 + 5.85i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.684 + 3.88i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (2.76 - 2.32i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.30 - 7.37i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (11.1 + 9.36i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.576 - 3.26i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-9.40 - 3.42i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-1.82 - 0.666i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.809 + 1.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (11.5 - 4.19i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (8.87 - 7.45i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.36234353351134024471012902479, −9.567941851365553572349091282629, −8.267573873179775629223748953352, −7.33421562438209699350437300356, −6.87653560165406652801346070590, −5.16592610156183269223549167822, −4.64307817846319069121327770426, −3.71103747997094422463534668554, −2.68899441440499040974881231891, −1.41028254984658104716543872644, 1.38757603183801067548350060453, 2.93147923144291178399021225470, 4.28709697720667658784568090144, 5.08871579165597684326346782625, 5.85291564043949820266587030533, 6.35571309716109190067827382220, 7.88645875893179410190011998684, 8.219920663600070914285667763584, 9.430911568545431701444668127156, 10.08414412791946950560630278844

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