↑

Properties

Label 2-855-15.2-c1-0-21
Degree $2$
Conductor $855$
Sign $0.670 + 0.742i$
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  + (1.47 − 1.47i)2-s − 2.33i·4-s + (−1.56 + 1.59i)5-s + (1.22 + 1.22i)7-s + (−0.492 − 0.492i)8-s + (0.0522 + 4.65i)10-s − 0.147i·11-s + (3.13 − 3.13i)13-s + 3.60·14-s + 3.21·16-s + (2.67 − 2.67i)17-s + i·19-s + (3.73 + 3.64i)20-s + (−0.216 − 0.216i)22-s + (5.66 + 5.66i)23-s + ⋯
L(s)  = 1  + (1.04 − 1.04i)2-s − 1.16i·4-s + (−0.699 + 0.714i)5-s + (0.463 + 0.463i)7-s + (−0.174 − 0.174i)8-s + (0.0165 + 1.47i)10-s − 0.0444i·11-s + (0.868 − 0.868i)13-s + 0.964·14-s + 0.804·16-s + (0.649 − 0.649i)17-s + 0.229i·19-s + (0.834 + 0.816i)20-s + (−0.0462 − 0.0462i)22-s + (1.18 + 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.670 + 0.742i$
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: \((2,\ 855,\ (\ :1/2),\ 0.670 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.49790 - 1.10963i\)
\(L(\frac12)\) \(\approx\) \(2.49790 - 1.10963i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.56 - 1.59i)T \)
19 \( 1 - iT \)
good2 \( 1 + (-1.47 + 1.47i)T - 2iT^{2} \)
7 \( 1 + (-1.22 - 1.22i)T + 7iT^{2} \)
11 \( 1 + 0.147iT - 11T^{2} \)
13 \( 1 + (-3.13 + 3.13i)T - 13iT^{2} \)
17 \( 1 + (-2.67 + 2.67i)T - 17iT^{2} \)
23 \( 1 + (-5.66 - 5.66i)T + 23iT^{2} \)
29 \( 1 + 4.33T + 29T^{2} \)
31 \( 1 - 9.06T + 31T^{2} \)
37 \( 1 + (1.36 + 1.36i)T + 37iT^{2} \)
41 \( 1 - 12.1iT - 41T^{2} \)
43 \( 1 + (0.718 - 0.718i)T - 43iT^{2} \)
47 \( 1 + (3.82 - 3.82i)T - 47iT^{2} \)
53 \( 1 + (6.09 + 6.09i)T + 53iT^{2} \)
59 \( 1 + 14.7T + 59T^{2} \)
61 \( 1 + 9.05T + 61T^{2} \)
67 \( 1 + (-3.19 - 3.19i)T + 67iT^{2} \)
71 \( 1 + 6.30iT - 71T^{2} \)
73 \( 1 + (1.43 - 1.43i)T - 73iT^{2} \)
79 \( 1 - 1.27iT - 79T^{2} \)
83 \( 1 + (3.60 + 3.60i)T + 83iT^{2} \)
89 \( 1 + 2.75T + 89T^{2} \)
97 \( 1 + (-1.79 - 1.79i)T + 97iT^{2} \)
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

−10.45754599118089290434675308513, −9.534206952256815067095338736869, −8.184564918493310744749577773298, −7.66619245021143934524642811365, −6.32276356827830485728754886980, −5.36701339196546649155048285332, −4.53317732983901572039115496646, −3.29951899725416633310617338815, −2.97734846056717855338830381735, −1.39153779116802029433108218267, 1.26303247171266050507046435928, 3.40560168095550865824815512673, 4.32718673154843608982389509431, 4.80988491566840268122834062522, 5.89764407168351285419854457209, 6.76571754098357811890057198836, 7.58060547058251696083487776071, 8.332496089058207870195093652570, 9.094416000915443528694310530824, 10.46016875151748610163565790619

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