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Properties

Label 2-855-19.7-c1-0-19
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  + (1.14 + 1.97i)2-s + (−1.61 + 2.79i)4-s + (−0.5 − 0.866i)5-s + 3.50·7-s − 2.79·8-s + (1.14 − 1.97i)10-s + 4.50·11-s + (2.5 − 4.33i)13-s + (4.00 + 6.94i)14-s + (0.0316 + 0.0547i)16-s + (0.0793 + 0.137i)17-s + (−4.26 − 0.920i)19-s + 3.22·20-s + (5.14 + 8.91i)22-s + (0.579 − 1.00i)23-s + ⋯
L(s)  = 1  + (0.807 + 1.39i)2-s + (−0.805 + 1.39i)4-s + (−0.223 − 0.387i)5-s + 1.32·7-s − 0.987·8-s + (0.361 − 0.625i)10-s + 1.35·11-s + (0.693 − 1.20i)13-s + (1.07 + 1.85i)14-s + (0.00790 + 0.0136i)16-s + (0.0192 + 0.0333i)17-s + (−0.977 − 0.211i)19-s + 0.720·20-s + (1.09 + 1.90i)22-s + (0.120 − 0.209i)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

\(L(1)\) \(\approx\) \(1.85499 + 2.04609i\)
\(L(\frac12)\) \(\approx\) \(1.85499 + 2.04609i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.26 + 0.920i)T \)
good2 \( 1 + (-1.14 - 1.97i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 - 3.50T + 7T^{2} \)
11 \( 1 - 4.50T + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.0793 - 0.137i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.579 + 1.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.75 - 3.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.28T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + (-3.03 - 5.26i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.67 - 2.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.53 + 2.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.87 - 4.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.53 - 2.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.436 + 0.756i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.22 + 7.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.11 + 14.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.57 - 6.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.06 - 8.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.85T + 83T^{2} \)
89 \( 1 + (0.556 - 0.963i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.809 + 1.40i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.58397754737383480373750546392, −9.020288027350677198385784275450, −8.435836517989934679562575119270, −7.79182665501498912104753063854, −6.87145421752408642689763502480, −5.99836827841205679684707179520, −5.15893207963018382639306160436, −4.40579250954443567283355052516, −3.55506685921991911433188923144, −1.47748021961720739342696918665, 1.44696843022817216828331123756, 2.14093075211920399384381005937, 3.76632719722354185379734238878, 4.11036052168669984667063970235, 5.13559357468523693885187638810, 6.30271380838113754286792217197, 7.31570119157122832581456215509, 8.564532177398196825210068669359, 9.238950611962673681696853742890, 10.39354919981124765160343514981

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