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Properties

Label 2-855-19.5-c1-0-7
Degree $2$
Conductor $855$
Sign $-0.781 - 0.623i$
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  + (−1.12 + 0.408i)2-s + (−0.439 + 0.369i)4-s + (0.766 + 0.642i)5-s + (1.09 + 1.90i)7-s + (1.53 − 2.66i)8-s + (−1.12 − 0.408i)10-s + (−1.41 + 2.44i)11-s + (0.708 + 4.01i)13-s + (−2.00 − 1.68i)14-s + (−0.437 + 2.48i)16-s + (0.359 − 0.130i)17-s + (−2.75 − 3.37i)19-s − 0.574·20-s + (0.586 − 3.32i)22-s + (2.27 − 1.91i)23-s + ⋯
L(s)  = 1  + (−0.793 + 0.288i)2-s + (−0.219 + 0.184i)4-s + (0.342 + 0.287i)5-s + (0.415 + 0.718i)7-s + (0.543 − 0.941i)8-s + (−0.354 − 0.129i)10-s + (−0.426 + 0.738i)11-s + (0.196 + 1.11i)13-s + (−0.536 − 0.450i)14-s + (−0.109 + 0.620i)16-s + (0.0872 − 0.0317i)17-s + (−0.632 − 0.774i)19-s − 0.128·20-s + (0.124 − 0.708i)22-s + (0.475 − 0.398i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.781 - 0.623i$
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: \((2,\ 855,\ (\ :1/2),\ -0.781 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259043 + 0.740483i\)
\(L(\frac12)\) \(\approx\) \(0.259043 + 0.740483i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.75 + 3.37i)T \)
good2 \( 1 + (1.12 - 0.408i)T + (1.53 - 1.28i)T^{2} \)
7 \( 1 + (-1.09 - 1.90i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.41 - 2.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.708 - 4.01i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.359 + 0.130i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.27 + 1.91i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-8.05 - 2.93i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.34 - 4.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + (0.544 - 3.08i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.53 - 1.29i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (10.2 + 3.72i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (5.72 - 4.80i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (12.1 - 4.43i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-4.57 + 3.84i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.986 + 0.358i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-5.99 - 5.03i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (2.44 - 13.8i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (2.33 - 13.2i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (2.77 + 4.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.13 - 6.46i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (13.5 - 4.94i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.28602220802764532841138618992, −9.536021392087122148900436689912, −8.705882722064544785483845100269, −8.291490490016891878402715862092, −6.97439403636433911917134201311, −6.65077700608070146206542451924, −5.11389337607574791784808302664, −4.40349514151568529916714546097, −2.90094342381719263795446938514, −1.62526432903522753053941966382, 0.52634198691151071176185594253, 1.68216627210905227192490287707, 3.18301518369421365648034116918, 4.55825896805483160250504287619, 5.38738372275826208530240587544, 6.30757470128597589465190669214, 7.78699429545210999328576766679, 8.179420431087920429799616291754, 8.993057643981072199619487609925, 10.03567402085022125531585084170

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