↑

Properties

Label 2-825-11.4-c1-0-29
Degree $2$
Conductor $825$
Sign $0.785 + 0.619i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
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  + (1.54 + 1.12i)2-s + (0.309 − 0.951i)3-s + (0.506 + 1.55i)4-s + (1.54 − 1.12i)6-s + (−1.37 − 4.23i)7-s + (0.213 − 0.656i)8-s + (−0.809 − 0.587i)9-s + (−1.22 + 3.08i)11-s + 1.63·12-s + (−0.432 − 0.313i)13-s + (2.62 − 8.08i)14-s + (3.71 − 2.69i)16-s + (3.67 − 2.67i)17-s + (−0.589 − 1.81i)18-s + (0.594 − 1.83i)19-s + ⋯
L(s)  = 1  + (1.09 + 0.792i)2-s + (0.178 − 0.549i)3-s + (0.253 + 0.779i)4-s + (0.629 − 0.457i)6-s + (−0.520 − 1.60i)7-s + (0.0753 − 0.231i)8-s + (−0.269 − 0.195i)9-s + (−0.368 + 0.929i)11-s + 0.472·12-s + (−0.119 − 0.0870i)13-s + (0.702 − 2.16i)14-s + (0.928 − 0.674i)16-s + (0.891 − 0.647i)17-s + (−0.138 − 0.427i)18-s + (0.136 − 0.419i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.785 + 0.619i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.785 + 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48997 - 0.863364i\)
\(L(\frac12)\) \(\approx\) \(2.48997 - 0.863364i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 \)
11 \( 1 + (1.22 - 3.08i)T \)
good2 \( 1 + (-1.54 - 1.12i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (1.37 + 4.23i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.432 + 0.313i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.67 + 2.67i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.594 + 1.83i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.18T + 23T^{2} \)
29 \( 1 + (1.46 + 4.52i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.82 + 2.05i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.162 - 0.501i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.45 + 4.46i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.02T + 43T^{2} \)
47 \( 1 + (3.70 - 11.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.06 - 2.95i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.21 - 12.9i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (9.45 - 6.87i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + (2.72 - 1.97i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.200 - 0.618i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-14.3 - 10.4i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.43 + 2.49i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 8.24T + 89T^{2} \)
97 \( 1 + (-0.920 - 0.669i)T + (29.9 + 92.2i)T^{2} \)
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

−10.05873521675124567247887110872, −9.424918037239332634385898802485, −7.81184605133814349977864536785, −7.27478513732231758686029221285, −6.85475307351203690383291792163, −5.75210786045235340721528589514, −4.76688158386599733774350994885, −3.94121324331542721503344808624, −2.89625105015649457369614234096, −0.938897240217945778556641165752, 2.00663805794076393969027779575, 3.17311405871891537273421801445, 3.50002210693355203565624548456, 5.11226707709350852371515094662, 5.42809893107952375028084168071, 6.39765905007425267409473008307, 8.060367526572713279284203215939, 8.711596110935513618584262962893, 9.588456614803247860761139674961, 10.54986245203523800555827580930

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