↑

Properties

Label 2-855-285.239-c0-0-2
Degree $2$
Conductor $855$
Sign $-0.462 + 0.886i$
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  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (−0.258 − 0.965i)5-s + i·7-s + (−1.36 − 0.366i)10-s − 1.41i·11-s + (0.866 − 0.5i)13-s + (1.22 + 0.707i)14-s + (0.499 − 0.866i)16-s + (−0.707 + 1.22i)17-s − 19-s + (−0.707 + 0.707i)20-s + (−1.73 − 1.00i)22-s + (−0.866 + 0.499i)25-s − 1.41i·26-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (−0.258 − 0.965i)5-s + i·7-s + (−1.36 − 0.366i)10-s − 1.41i·11-s + (0.866 − 0.5i)13-s + (1.22 + 0.707i)14-s + (0.499 − 0.866i)16-s + (−0.707 + 1.22i)17-s − 19-s + (−0.707 + 0.707i)20-s + (−1.73 − 1.00i)22-s + (−0.866 + 0.499i)25-s − 1.41i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.462 + 0.886i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ -0.462 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.383107639\)
\(L(\frac12)\) \(\approx\) \(1.383107639\)
\(L(1)\) 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 \( 1 + (0.258 + 0.965i)T \)
19 \( 1 + T \)
good2 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−10.57640087683351708628308881551, −9.186635031040422838159402101626, −8.622882330292783908597263057521, −7.935478970052452293694108822964, −6.09839571544282348243685301067, −5.61886401624823772075762263516, −4.42543192042194681394949982803, −3.68035661184209559911763535399, −2.58863311450106908323600675115, −1.33352474806981839303456643594, 2.19635831488147365476826230655, 3.97058395931699958610031398542, 4.25625885401448125634083548471, 5.52042023216544637029382603249, 6.62619241857471946087747866764, 7.11110886170110090324142813627, 7.51792598579654628638700480190, 8.744247438800929580831956570958, 9.864085175324200494650777461107, 10.75581186687532635425174238404

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