↑

Properties

Label 2-855-19.6-c1-0-24
Degree $2$
Conductor $855$
Sign $0.501 + 0.865i$
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  + (−0.156 + 0.130i)2-s + (−0.340 + 1.92i)4-s + (−0.173 − 0.984i)5-s + (0.386 − 0.670i)7-s + (−0.403 − 0.698i)8-s + (0.156 + 0.130i)10-s + (−2.05 − 3.56i)11-s + (−2.36 + 0.859i)13-s + (0.0273 + 0.155i)14-s + (−3.52 − 1.28i)16-s + (4.87 − 4.09i)17-s + (3.29 − 2.85i)19-s + 1.95·20-s + (0.787 + 0.286i)22-s + (0.0301 − 0.171i)23-s + ⋯
L(s)  = 1  + (−0.110 + 0.0925i)2-s + (−0.170 + 0.964i)4-s + (−0.0776 − 0.440i)5-s + (0.146 − 0.253i)7-s + (−0.142 − 0.246i)8-s + (0.0493 + 0.0414i)10-s + (−0.620 − 1.07i)11-s + (−0.655 + 0.238i)13-s + (0.00731 + 0.0414i)14-s + (−0.881 − 0.320i)16-s + (1.18 − 0.992i)17-s + (0.755 − 0.655i)19-s + 0.437·20-s + (0.167 + 0.0611i)22-s + (0.00628 − 0.0356i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.501 + 0.865i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (766, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.501 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.942653 - 0.543258i\)
\(L(\frac12)\) \(\approx\) \(0.942653 - 0.543258i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-3.29 + 2.85i)T \)
good2 \( 1 + (0.156 - 0.130i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (-0.386 + 0.670i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.05 + 3.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.36 - 0.859i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-4.87 + 4.09i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.0301 + 0.171i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.22 + 2.70i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-3.40 + 5.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.25T + 37T^{2} \)
41 \( 1 + (-11.6 - 4.25i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.90 - 10.7i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (6.02 + 5.05i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-2.12 + 12.0i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-6.08 + 5.10i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.55 + 8.79i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (10.8 + 9.10i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.302 + 1.71i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-11.4 - 4.18i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-6.27 - 2.28i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-4.99 + 8.65i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (13.4 - 4.88i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.168 + 0.141i)T + (16.8 - 95.5i)T^{2} \)
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

−9.701327259606040253398743517809, −9.297293051055503844729787084310, −7.973458274985978842241849428634, −7.88326205775154388334173992566, −6.78348478242064943027899793073, −5.48813641042584400687459555967, −4.68758912188258100996784156452, −3.51925547245632587971802334921, −2.64182092290381336790918240749, −0.57575898775009894053097800794, 1.46693457966173675327939087180, 2.62924839638498295611192293666, 4.01849089406705851328037696331, 5.29045305215413928278476575591, 5.67174458364333707253588673112, 6.99489973278964135365424090117, 7.66407619652390460135541307835, 8.748982379441365568589302838127, 9.721324898037798014534657831257, 10.33480114538874908325183249034

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