↑

Properties

Label 2-855-19.7-c1-0-3
Degree $2$
Conductor $855$
Sign $0.282 - 0.959i$
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  + (−0.595 − 1.03i)2-s + (0.290 − 0.503i)4-s + (0.5 + 0.866i)5-s − 0.609·7-s − 3.07·8-s + (0.595 − 1.03i)10-s − 4.48·11-s + (−2.21 + 3.84i)13-s + (0.362 + 0.628i)14-s + (1.24 + 2.16i)16-s + (1.45 + 2.51i)17-s + (3.60 + 2.44i)19-s + 0.581·20-s + (2.67 + 4.62i)22-s + (−1.42 + 2.46i)23-s + ⋯
L(s)  = 1  + (−0.421 − 0.729i)2-s + (0.145 − 0.251i)4-s + (0.223 + 0.387i)5-s − 0.230·7-s − 1.08·8-s + (0.188 − 0.326i)10-s − 1.35·11-s + (−0.615 + 1.06i)13-s + (0.0969 + 0.167i)14-s + (0.312 + 0.540i)16-s + (0.352 + 0.609i)17-s + (0.827 + 0.562i)19-s + 0.130·20-s + (0.569 + 0.986i)22-s + (−0.296 + 0.514i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.282 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.282 - 0.959i$
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: \((2,\ 855,\ (\ :1/2),\ 0.282 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420116 + 0.314333i\)
\(L(\frac12)\) \(\approx\) \(0.420116 + 0.314333i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 - 2.44i)T \)
good2 \( 1 + (0.595 + 1.03i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 0.609T + 7T^{2} \)
11 \( 1 + 4.48T + 11T^{2} \)
13 \( 1 + (2.21 - 3.84i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.45 - 2.51i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.42 - 2.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.558 + 0.966i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.22T + 31T^{2} \)
37 \( 1 + 3.77T + 37T^{2} \)
41 \( 1 + (4.15 + 7.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.99 - 8.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.94 - 5.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.22 + 7.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.11 - 8.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.49 + 4.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.23 - 7.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.80 - 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.86 + 3.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.51 + 7.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.12T + 83T^{2} \)
89 \( 1 + (-3.96 + 6.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.83 - 8.37i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.20384414117166767545166489681, −9.831355071483750068126506829709, −8.958110059469699145665970769097, −7.83682689963285029899443427814, −6.97654491242863026353974788926, −5.92499875672587147782804030108, −5.19702707644229003855813793613, −3.65332191341861130355206876142, −2.61609443620344090910298769722, −1.66323743063810497888715709465, 0.27068940705719727939309326694, 2.50306898755738545133927672406, 3.34460127543528603200185074033, 5.09335044298263131730944351380, 5.54013719001459856676738810036, 6.77278597749875741362923133694, 7.57660599147416606957597675585, 8.108219656281114970349708119707, 9.043010222951174648494496062619, 9.853809885344265773382771922309

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