↑

Properties

Label 2-855-95.59-c0-0-0
Degree $2$
Conductor $855$
Sign $-0.150 - 0.988i$
Analytic cond. $0.426700$
Root an. cond. $0.653223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.118 + 0.673i)2-s + (0.500 + 0.181i)4-s + (0.342 + 0.939i)5-s + (−0.524 + 0.907i)8-s + (−0.673 + 0.118i)10-s + (−0.141 − 0.118i)16-s + (−0.342 − 0.0603i)17-s + (−0.766 − 0.642i)19-s + 0.532i·20-s + (0.342 − 0.939i)23-s + (−0.766 + 0.642i)25-s + (1.11 − 0.642i)31-s + (−0.705 + 0.592i)32-s + (0.0812 − 0.223i)34-s + (0.524 − 0.439i)38-s + ⋯
L(s)  = 1  + (−0.118 + 0.673i)2-s + (0.500 + 0.181i)4-s + (0.342 + 0.939i)5-s + (−0.524 + 0.907i)8-s + (−0.673 + 0.118i)10-s + (−0.141 − 0.118i)16-s + (−0.342 − 0.0603i)17-s + (−0.766 − 0.642i)19-s + 0.532i·20-s + (0.342 − 0.939i)23-s + (−0.766 + 0.642i)25-s + (1.11 − 0.642i)31-s + (−0.705 + 0.592i)32-s + (0.0812 − 0.223i)34-s + (0.524 − 0.439i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.150 - 0.988i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ -0.150 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.095702549\)
\(L(\frac12)\) \(\approx\) \(1.095702549\)
\(L(1)\) 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 \( 1 + (-0.342 - 0.939i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
good2 \( 1 + (0.118 - 0.673i)T + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.342 + 0.0603i)T + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.342 + 0.939i)T + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (-1.85 + 0.326i)T + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (1.85 + 0.673i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.173 - 0.984i)T^{2} \)
79 \( 1 + (1.11 - 1.32i)T + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-1.32 + 0.766i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.939 - 0.342i)T^{2} \)
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

−10.76945367561140234198404365727, −9.762480865938295908686819524263, −8.750301123390331694215398519454, −7.947136080113666765541826987627, −6.99517266420022984794195162819, −6.50545763106275688793379886117, −5.71252965043180046986763369684, −4.42909330413909024108702918398, −2.98944711813059359643176094028, −2.22359030686451078590105993652, 1.27569170227014769596321539895, 2.32265485201821239871860641778, 3.61044326667727460623544865386, 4.73468670628537166533734246744, 5.81376339322624795883612369055, 6.56557766538673142202409624662, 7.70419232506131210994456110478, 8.686236488134307098982162071019, 9.451032462264560939485532597475, 10.19086930041725523581867217662

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