↑

Properties

Label 2-855-15.2-c1-0-10
Degree $2$
Conductor $855$
Sign $0.374 - 0.927i$
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  + (0.707 − 0.707i)2-s + 0.999i·4-s + (−2.12 − 0.707i)5-s + (3 + 3i)7-s + (2.12 + 2.12i)8-s + (−2 + 0.999i)10-s − 4.24i·11-s + (−4 + 4i)13-s + 4.24·14-s + 1.00·16-s + (−4.24 + 4.24i)17-s − i·19-s + (0.707 − 2.12i)20-s + (−3 − 3i)22-s + (4.24 + 4.24i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + 0.499i·4-s + (−0.948 − 0.316i)5-s + (1.13 + 1.13i)7-s + (0.750 + 0.750i)8-s + (−0.632 + 0.316i)10-s − 1.27i·11-s + (−1.10 + 1.10i)13-s + 1.13·14-s + 0.250·16-s + (−1.02 + 1.02i)17-s − 0.229i·19-s + (0.158 − 0.474i)20-s + (−0.639 − 0.639i)22-s + (0.884 + 0.884i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35176 + 0.912088i\)
\(L(\frac12)\) \(\approx\) \(1.35176 + 0.912088i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 + 0.707i)T \)
19 \( 1 + iT \)
good2 \( 1 + (-0.707 + 0.707i)T - 2iT^{2} \)
7 \( 1 + (-3 - 3i)T + 7iT^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 + (4 - 4i)T - 13iT^{2} \)
17 \( 1 + (4.24 - 4.24i)T - 17iT^{2} \)
23 \( 1 + (-4.24 - 4.24i)T + 23iT^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-6 - 6i)T + 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-7 + 7i)T - 43iT^{2} \)
47 \( 1 + (-1.41 + 1.41i)T - 47iT^{2} \)
53 \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (6 + 6i)T + 67iT^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 - 8iT - 79T^{2} \)
83 \( 1 + (2.82 + 2.82i)T + 83iT^{2} \)
89 \( 1 + 5.65T + 89T^{2} \)
97 \( 1 + (6 + 6i)T + 97iT^{2} \)
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

−10.93733088069125050099740964831, −9.129711650936315421097105331894, −8.660675270167157803499311093034, −7.948658859435324802207955235461, −7.09631985179969891976800271923, −5.64768599686169725891334034308, −4.75648528966673451467795539197, −4.06898520767724501160275018494, −2.89959657532115704900282928127, −1.82422050502618107664821739428, 0.68339831818531473134209015564, 2.39410476387501474313726119822, 4.12995728373954615154208157743, 4.60434725871545345097120026377, 5.34137476641225305486381392827, 6.93130029262626244957050539370, 7.30216170741332012981113820070, 7.83549012913113062480177926419, 9.232674689803412727319243874242, 10.26339102904403426568611085140

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