Properties

Label 2-855-95.4-c1-0-1
Degree $2$
Conductor $855$
Sign $-0.449 + 0.893i$
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  + (−0.795 + 2.18i)2-s + (−2.60 − 2.18i)4-s + (0.914 − 2.04i)5-s + (0.124 + 0.0716i)7-s + (2.82 − 1.63i)8-s + (3.73 + 3.61i)10-s + (−1.40 − 2.43i)11-s + (−1.68 − 0.296i)13-s + (−0.255 + 0.214i)14-s + (0.135 + 0.766i)16-s + (−1.21 + 3.34i)17-s + (−2.82 + 3.32i)19-s + (−6.84 + 3.32i)20-s + (6.44 − 1.13i)22-s + (−4.62 + 5.51i)23-s + ⋯
L(s)  = 1  + (−0.562 + 1.54i)2-s + (−1.30 − 1.09i)4-s + (0.408 − 0.912i)5-s + (0.0469 + 0.0270i)7-s + (0.999 − 0.576i)8-s + (1.17 + 1.14i)10-s + (−0.424 − 0.735i)11-s + (−0.466 − 0.0822i)13-s + (−0.0682 + 0.0572i)14-s + (0.0338 + 0.191i)16-s + (−0.295 + 0.811i)17-s + (−0.647 + 0.762i)19-s + (−1.53 + 0.742i)20-s + (1.37 − 0.242i)22-s + (−0.964 + 1.14i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.449 + 0.893i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0731811 - 0.118702i\)
\(L(\frac12)\) \(\approx\) \(0.0731811 - 0.118702i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.914 + 2.04i)T \)
19 \( 1 + (2.82 - 3.32i)T \)
good2 \( 1 + (0.795 - 2.18i)T + (-1.53 - 1.28i)T^{2} \)
7 \( 1 + (-0.124 - 0.0716i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.40 + 2.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.68 + 0.296i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.21 - 3.34i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (4.62 - 5.51i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (7.11 - 2.58i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.42 - 4.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.49iT - 37T^{2} \)
41 \( 1 + (-0.0325 - 0.184i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (7.06 + 8.42i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (1.25 + 3.44i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (1.05 - 1.25i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-5.04 - 1.83i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-7.74 - 6.50i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.05 + 5.64i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (0.414 - 0.347i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-10.3 + 1.81i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-2.01 - 11.4i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (6.57 + 3.79i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.251 + 1.42i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-3.01 + 8.27i)T + (-74.3 - 62.3i)T^{2} \)
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

−10.24371250444419693362176291411, −9.701019646866166470689895675697, −8.558864175031258419158040040107, −8.416982758401389949049611268850, −7.43837085001615732940454810871, −6.41700559139640702354961108716, −5.60974937277649368608900997948, −5.10144433197304279347595737213, −3.75027850034928810134937358358, −1.74739008702482819041275254290, 0.07633004245969081547360619271, 2.12214199016724763941659841636, 2.49327185183415519624858164594, 3.77290405893282914817121144240, 4.78140814886297717377370197587, 6.19020281310055448976692525769, 7.20352612260931751202083811922, 8.095828026110984460486569079980, 9.339906237355539126751232405211, 9.678383548879008379830489659762

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