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Properties

Label 2-855-19.7-c1-0-1
Degree $2$
Conductor $855$
Sign $-0.0977 + 0.995i$
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 + 2.09i)2-s + (−1.91 + 3.31i)4-s + (−0.5 − 0.866i)5-s − 3.82·7-s − 4.41·8-s + (1.20 − 2.09i)10-s − 2.82·11-s + (−1.91 + 3.31i)13-s + (−4.62 − 8.00i)14-s + (−1.49 − 2.59i)16-s + (−3.41 − 5.91i)17-s + (4 + 1.73i)19-s + 3.82·20-s + (−3.41 − 5.91i)22-s + (2.41 − 4.18i)23-s + ⋯
L(s)  = 1  + (0.853 + 1.47i)2-s + (−0.957 + 1.65i)4-s + (−0.223 − 0.387i)5-s − 1.44·7-s − 1.56·8-s + (0.381 − 0.661i)10-s − 0.852·11-s + (−0.530 + 0.919i)13-s + (−1.23 − 2.13i)14-s + (−0.374 − 0.649i)16-s + (−0.828 − 1.43i)17-s + (0.917 + 0.397i)19-s + 0.856·20-s + (−0.727 − 1.26i)22-s + (0.503 − 0.871i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.0977 + 0.995i$
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: \((2,\ 855,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278820 - 0.307543i\)
\(L(\frac12)\) \(\approx\) \(0.278820 - 0.307543i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 - 1.73i)T \)
good2 \( 1 + (-1.20 - 2.09i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 3.82T + 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + (1.91 - 3.31i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.41 + 5.91i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.41 + 4.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.828 - 1.43i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 7.82T + 37T^{2} \)
41 \( 1 + (1.41 + 2.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.08 + 1.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.41 - 7.64i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.15 - 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.74 + 9.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5 - 8.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.74 - 6.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.32 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (2.24 - 3.88i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.77768192069703597719020214192, −9.529725391583880066031135354326, −9.046892409740634023086598909624, −7.933425913025809496941666197780, −6.98427089782899625206182493341, −6.72688556460919425717174805658, −5.47604792205812285833211971385, −4.89282408308823625802978593287, −3.83131101300964818251869368282, −2.77931632868215756296614681487, 0.14091225595846314830869586172, 2.05070944851746593685893097010, 3.25065695249925714829790988553, 3.47933169873033164158730829236, 4.91284907153272873165746223580, 5.71111025177151462283641598065, 6.78177968712170771112649359570, 7.85655746134008778644626033773, 9.194408617558616987773922792443, 9.949191021937475986026007195661

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