↑

Properties

Label 2-1638-91.34-c1-0-34
Degree $2$
Conductor $1638$
Sign $-0.536 + 0.844i$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.864 − 0.864i)5-s + (−1.70 + 2.02i)7-s + (0.707 − 0.707i)8-s − 1.22·10-s + (3.50 − 3.50i)11-s + (−3.37 − 1.25i)13-s + (2.63 − 0.222i)14-s − 1.00·16-s − 0.322·17-s + (1.77 − 1.77i)19-s + (0.864 + 0.864i)20-s − 4.95·22-s − 2.70i·23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.386 − 0.386i)5-s + (−0.645 + 0.763i)7-s + (0.250 − 0.250i)8-s − 0.386·10-s + (1.05 − 1.05i)11-s + (−0.937 − 0.348i)13-s + (0.704 − 0.0593i)14-s − 0.250·16-s − 0.0781·17-s + (0.406 − 0.406i)19-s + (0.193 + 0.193i)20-s − 1.05·22-s − 0.564i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.536 + 0.844i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.536 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9680603655\)
\(L(\frac12)\) \(\approx\) \(0.9680603655\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (1.70 - 2.02i)T \)
13 \( 1 + (3.37 + 1.25i)T \)
good5 \( 1 + (-0.864 + 0.864i)T - 5iT^{2} \)
11 \( 1 + (-3.50 + 3.50i)T - 11iT^{2} \)
17 \( 1 + 0.322T + 17T^{2} \)
19 \( 1 + (-1.77 + 1.77i)T - 19iT^{2} \)
23 \( 1 + 2.70iT - 23T^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 + (1.72 - 1.72i)T - 31iT^{2} \)
37 \( 1 + (-2.27 + 2.27i)T - 37iT^{2} \)
41 \( 1 + (-6.46 + 6.46i)T - 41iT^{2} \)
43 \( 1 + 0.393iT - 43T^{2} \)
47 \( 1 + (1.41 + 1.41i)T + 47iT^{2} \)
53 \( 1 - 2.03T + 53T^{2} \)
59 \( 1 + (10.7 + 10.7i)T + 59iT^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 + (-5.38 - 5.38i)T + 67iT^{2} \)
71 \( 1 + (-0.647 - 0.647i)T + 71iT^{2} \)
73 \( 1 + (6.85 + 6.85i)T + 73iT^{2} \)
79 \( 1 - 8.81T + 79T^{2} \)
83 \( 1 + (-6.09 + 6.09i)T - 83iT^{2} \)
89 \( 1 + (3.68 + 3.68i)T + 89iT^{2} \)
97 \( 1 + (-5.20 + 5.20i)T - 97iT^{2} \)
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

−9.332576417031532498176606014458, −8.639151923671635934520098772164, −7.65506586028965749749642109152, −6.70917749825186480271254969777, −5.85624437714115406957199868119, −5.06477939268097809955765444763, −3.75913795397159205496959590867, −2.93535781916019026330856780948, −1.87265420368278072195789725046, −0.45868108599408485358168833652, 1.32542852485837861337782298768, 2.52583903052832469228487886699, 3.88555367365399986490565600418, 4.66257356251345975779840752912, 5.89175146314817188710573348565, 6.59387925021397646994616097508, 7.26638322743820029839359000883, 7.78010364354208623280992900657, 9.151146142087183354691426842685, 9.626757646294933527722716188410

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