Properties

Label 2-855-19.7-c1-0-14
Degree $2$
Conductor $855$
Sign $-0.813 + 0.582i$
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  + (−1.25 − 2.17i)2-s + (−2.14 + 3.71i)4-s + (−0.5 − 0.866i)5-s − 0.221·7-s + 5.72·8-s + (−1.25 + 2.17i)10-s + 0.778·11-s + (2.5 − 4.33i)13-s + (0.278 + 0.481i)14-s + (−2.89 − 5.01i)16-s + (3.53 + 6.12i)17-s + (1.33 + 4.15i)19-s + 4.28·20-s + (−0.975 − 1.68i)22-s + (4.03 − 6.99i)23-s + ⋯
L(s)  = 1  + (−0.886 − 1.53i)2-s + (−1.07 + 1.85i)4-s + (−0.223 − 0.387i)5-s − 0.0838·7-s + 2.02·8-s + (−0.396 + 0.686i)10-s + 0.234·11-s + (0.693 − 1.20i)13-s + (0.0743 + 0.128i)14-s + (−0.724 − 1.25i)16-s + (0.858 + 1.48i)17-s + (0.305 + 0.952i)19-s + 0.958·20-s + (−0.207 − 0.360i)22-s + (0.842 − 1.45i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.813 + 0.582i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.263141 - 0.819373i\)
\(L(\frac12)\) \(\approx\) \(0.263141 - 0.819373i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 - 4.15i)T \)
good2 \( 1 + (1.25 + 2.17i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 0.221T + 7T^{2} \)
11 \( 1 - 0.778T + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.53 - 6.12i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.03 + 6.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.110 + 0.192i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 + 1.90T + 37T^{2} \)
41 \( 1 + (3.61 + 6.26i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.64 + 6.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.39 - 2.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.19 + 3.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.39 + 2.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.29 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.28 + 9.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.92 + 8.52i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.03 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.792 + 1.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.52T + 83T^{2} \)
89 \( 1 + (-1.57 + 2.71i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.18 - 5.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.19840043327876221808514081397, −9.099204150865725331147498206569, −8.251802364272967602692057660928, −8.006144619620775601142410160153, −6.41494137843336007702231767322, −5.19207595269564458102058354266, −3.80721491750881589511302228519, −3.28139273191288228166320499463, −1.83195136494491115977167745430, −0.72094563299972330615224213059, 1.14564056617295789954246641625, 3.17455059096047881226755744440, 4.66272543541607611111443837977, 5.50854718627252836350811122878, 6.60117903658806493246435335726, 7.06163146215140943796205668627, 7.80621779225656142920429052226, 8.815537272026923827648681741600, 9.409236850473198794405353817714, 10.02353205639661301890981016913

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