Properties

Label 2-1638-91.81-c1-0-36
Degree $2$
Conductor $1638$
Sign $0.0710 + 0.997i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.228 + 0.395i)5-s + (2.45 − 0.989i)7-s − 0.999·8-s + 0.456·10-s + 3.83·11-s + (−3.13 − 1.78i)13-s + (0.369 − 2.61i)14-s + (−0.5 + 0.866i)16-s + (0.775 + 1.34i)17-s − 2.88·19-s + (0.228 − 0.395i)20-s + (1.91 − 3.32i)22-s + (1.62 − 2.80i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.102 + 0.176i)5-s + (0.927 − 0.374i)7-s − 0.353·8-s + 0.144·10-s + 1.15·11-s + (−0.869 − 0.494i)13-s + (0.0988 − 0.700i)14-s + (−0.125 + 0.216i)16-s + (0.188 + 0.325i)17-s − 0.661·19-s + (0.0510 − 0.0883i)20-s + (0.409 − 0.708i)22-s + (0.338 − 0.585i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0710 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0710 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.0710 + 0.997i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.0710 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.301174445\)
\(L(\frac12)\) \(\approx\) \(2.301174445\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-2.45 + 0.989i)T \)
13 \( 1 + (3.13 + 1.78i)T \)
good5 \( 1 + (-0.228 - 0.395i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.83T + 11T^{2} \)
17 \( 1 + (-0.775 - 1.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.88T + 19T^{2} \)
23 \( 1 + (-1.62 + 2.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.20 - 3.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.80 + 8.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.140 - 0.242i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.57 - 6.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.21 + 2.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.93 - 6.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.550 - 0.953i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.68 + 8.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 + 1.78T + 67T^{2} \)
71 \( 1 + (-5.06 + 8.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.40 + 2.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.70 - 4.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.35T + 83T^{2} \)
89 \( 1 + (0.0898 - 0.155i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.73 + 8.20i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.330610246373908575479007727023, −8.438592392286133739034852275019, −7.68483133269106960597400198814, −6.64710846429361038366404522644, −5.92275944662662366645000844083, −4.65652673911216309415762469041, −4.35966299371293504370156395501, −3.08314149127824459320673045548, −2.07126156185774014372992943190, −0.894450758319033610378501108969, 1.36706633605097688247346449056, 2.62103945657343960517990458527, 3.92449994618610155107089706096, 4.73759966657236023442327399564, 5.38005437329958252108723863096, 6.39744229577495683814971093473, 7.12390461215774974655400304543, 7.87437533819850901594819439302, 8.917785689823577964284531588173, 9.136713029765048025345088745721

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