↑

Properties

Label 2-855-19.9-c1-0-30
Degree $2$
Conductor $855$
Sign $0.123 + 0.992i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0366 − 0.207i)2-s + (1.83 − 0.668i)4-s + (0.939 + 0.342i)5-s + (0.843 − 1.46i)7-s + (−0.417 − 0.722i)8-s + (0.0366 − 0.207i)10-s + (−1.44 − 2.50i)11-s + (−4.95 − 4.15i)13-s + (−0.334 − 0.121i)14-s + (2.86 − 2.40i)16-s + (−0.518 − 2.94i)17-s + (−4.34 − 0.288i)19-s + 1.95·20-s + (−0.466 + 0.391i)22-s + (7.75 − 2.82i)23-s + ⋯
L(s)  = 1  + (−0.0259 − 0.146i)2-s + (0.918 − 0.334i)4-s + (0.420 + 0.152i)5-s + (0.318 − 0.552i)7-s + (−0.147 − 0.255i)8-s + (0.0115 − 0.0657i)10-s + (−0.435 − 0.753i)11-s + (−1.37 − 1.15i)13-s + (−0.0894 − 0.0325i)14-s + (0.715 − 0.600i)16-s + (−0.125 − 0.713i)17-s + (−0.997 − 0.0662i)19-s + 0.437·20-s + (−0.0994 + 0.0834i)22-s + (1.61 − 0.588i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.123 + 0.992i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.123 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39687 - 1.23349i\)
\(L(\frac12)\) \(\approx\) \(1.39687 - 1.23349i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (4.34 + 0.288i)T \)
good2 \( 1 + (0.0366 + 0.207i)T + (-1.87 + 0.684i)T^{2} \)
7 \( 1 + (-0.843 + 1.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.44 + 2.50i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.95 + 4.15i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.518 + 2.94i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-7.75 + 2.82i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.26 - 7.14i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.02 - 3.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.96T + 37T^{2} \)
41 \( 1 + (4.17 - 3.50i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-5.01 - 1.82i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.286 - 1.62i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-1.79 + 0.653i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.616 + 3.49i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-7.42 + 2.70i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.393 + 2.23i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-9.79 - 3.56i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.08 + 0.907i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (1.84 - 1.54i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (5.70 - 9.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.74 - 6.50i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.71 + 9.71i)T + (-91.1 + 33.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.20935761576004713790766882983, −9.321228266088320040615581784258, −8.172225588829286195374229324519, −7.28096272303707745583124016447, −6.67320461889043961212307025466, −5.52302468904037605375847752658, −4.84725364216167131317279296385, −3.12433871647861142287653389531, −2.45984018619187888394287120784, −0.877725741325698047035025351997, 2.01251226084123665004007327830, 2.44994299002735781688875080724, 4.12493846234270117575392535287, 5.14603323054955523435893044803, 6.11430773427324355963578610039, 7.01908854511191441118155861762, 7.65788198759882009551596395387, 8.688439756234882660538257902057, 9.511649837219559407496826718420, 10.36156542532922173263489678892

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