Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.737 + 0.675i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 − 0.418i)2-s − 3-s + (1.64 + 1.13i)4-s − 1.44i·5-s + (1.35 + 0.418i)6-s + 2.69·7-s + (−1.75 − 2.21i)8-s + 9-s + (−0.606 + 1.95i)10-s − 3.96·11-s + (−1.64 − 1.13i)12-s − 2.95i·13-s + (−3.64 − 1.13i)14-s + 1.44i·15-s + (1.44 + 3.73i)16-s − 2.16·17-s + ⋯
L(s)  = 1  + (−0.955 − 0.296i)2-s − 0.577·3-s + (0.824 + 0.565i)4-s − 0.647i·5-s + (0.551 + 0.170i)6-s + 1.01·7-s + (−0.620 − 0.784i)8-s + 0.333·9-s + (−0.191 + 0.618i)10-s − 1.19·11-s + (−0.476 − 0.326i)12-s − 0.818i·13-s + (−0.974 − 0.302i)14-s + 0.374i·15-s + (0.360 + 0.932i)16-s − 0.524·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.737 + 0.675i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.737 + 0.675i)$
$L(1)$  $\approx$  $0.205018 - 0.527438i$
$L(\frac12)$  $\approx$  $0.205018 - 0.527438i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;67\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.418i)T \)
3 \( 1 + T \)
67 \( 1 + (-1.85 + 7.97i)T \)
good5 \( 1 + 1.44iT - 5T^{2} \)
7 \( 1 - 2.69T + 7T^{2} \)
11 \( 1 + 3.96T + 11T^{2} \)
13 \( 1 + 2.95iT - 13T^{2} \)
17 \( 1 + 2.16T + 17T^{2} \)
19 \( 1 + 6.52iT - 19T^{2} \)
23 \( 1 - 5.27iT - 23T^{2} \)
29 \( 1 - 1.48T + 29T^{2} \)
31 \( 1 - 4.41T + 31T^{2} \)
37 \( 1 + 6.67T + 37T^{2} \)
41 \( 1 + 0.555iT - 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 - 3.34iT - 53T^{2} \)
59 \( 1 + 7.97iT - 59T^{2} \)
61 \( 1 + 0.843iT - 61T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + 8.46T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + 8.13iT - 83T^{2} \)
89 \( 1 - 3.50T + 89T^{2} \)
97 \( 1 + 8.55iT - 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.10817960781206889507243645994, −8.991123690737127563517166448500, −8.311912246608107032718252569752, −7.59556911882753069120084379671, −6.67414492441839893268360151866, −5.31336577035385040106544929594, −4.78077320339661924683485840402, −3.09337230225823253686009372571, −1.78128457264328723793268334122, −0.41396006874749833423698679929, 1.55749612124157669073659715453, 2.71360509939890404162212860869, 4.51376401517442611238490176895, 5.46293168839525881167438981054, 6.44681352939478417925182997884, 7.16004361268143756084612310881, 8.114664840851884094434141278683, 8.650176154409767147317020383987, 10.07995391644510597196069892871, 10.40087122138218520999172224948

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