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Properties

Label 2-855-9.4-c1-0-2
Degree $2$
Conductor $855$
Sign $0.988 - 0.154i$
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  + (−1.04 − 1.80i)2-s + (−0.431 − 1.67i)3-s + (−1.16 + 2.01i)4-s + (0.5 − 0.866i)5-s + (−2.57 + 2.52i)6-s + (1.56 + 2.71i)7-s + 0.684·8-s + (−2.62 + 1.44i)9-s − 2.08·10-s + (−1.70 − 2.95i)11-s + (3.88 + 1.08i)12-s + (−1.25 + 2.17i)13-s + (3.26 − 5.65i)14-s + (−1.66 − 0.464i)15-s + (1.61 + 2.80i)16-s − 5.45·17-s + ⋯
L(s)  = 1  + (−0.735 − 1.27i)2-s + (−0.249 − 0.968i)3-s + (−0.582 + 1.00i)4-s + (0.223 − 0.387i)5-s + (−1.05 + 1.03i)6-s + (0.592 + 1.02i)7-s + 0.242·8-s + (−0.875 + 0.483i)9-s − 0.657·10-s + (−0.514 − 0.890i)11-s + (1.12 + 0.312i)12-s + (−0.348 + 0.604i)13-s + (0.871 − 1.51i)14-s + (−0.430 − 0.119i)15-s + (0.404 + 0.700i)16-s − 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.988 - 0.154i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.988 - 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.314886 + 0.0244549i\)
\(L(\frac12)\) \(\approx\) \(0.314886 + 0.0244549i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.431 + 1.67i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 - T \)
good2 \( 1 + (1.04 + 1.80i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.56 - 2.71i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.25 - 2.17i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.45T + 17T^{2} \)
23 \( 1 + (3.03 - 5.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.50 - 7.80i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.88 - 6.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.69T + 37T^{2} \)
41 \( 1 + (1.32 - 2.28i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.87 + 3.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.23 + 9.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.54T + 53T^{2} \)
59 \( 1 + (1.95 - 3.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.66 - 2.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.16 - 2.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + (5.43 + 9.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.39 - 4.15i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.98T + 89T^{2} \)
97 \( 1 + (7.72 + 13.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.38848391451348542592534292951, −9.189195093491956210631478820984, −8.694594257304337492600002678268, −8.102822800454978209041997615130, −6.82607161283158465372659020255, −5.77964942879439428205452550725, −4.96264863401902847664219702670, −3.19671872734903225640482362172, −2.19397707040476513173679499831, −1.45744406950678101235979820348, 0.19896497541328986925544444289, 2.59709141497516612182533439540, 4.24487839907833126825560980893, 4.87027609926737233646583399005, 6.02884399294261173955699388379, 6.72177375477573873708921597491, 7.77312644602342885607855432404, 8.186129107970782555875910293721, 9.468601164504300817852839733674, 9.885265370692542815916364069009

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