Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.920 - 0.389i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 + 1.13i)3-s + (2.17 + 0.312i)7-s + (0.426 − 2.96i)9-s + (2.69 − 1.23i)13-s + (−0.372 − 2.59i)19-s + (−3.20 + 2.05i)21-s + (−2.07 − 4.54i)25-s + (2.80 + 4.37i)27-s + (9.91 + 4.53i)31-s + 10.4·37-s + (−2.13 + 4.67i)39-s + (3.26 + 11.1i)43-s + (−2.08 − 0.612i)49-s + (3.43 + 2.97i)57-s + (3.26 + 5.08i)61-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)3-s + (0.821 + 0.118i)7-s + (0.142 − 0.989i)9-s + (0.748 − 0.341i)13-s + (−0.0855 − 0.595i)19-s + (−0.698 + 0.448i)21-s + (−0.415 − 0.909i)25-s + (0.540 + 0.841i)27-s + (1.78 + 0.813i)31-s + 1.71·37-s + (−0.341 + 0.748i)39-s + (0.498 + 1.69i)43-s + (−0.297 − 0.0874i)49-s + (0.454 + 0.393i)57-s + (0.418 + 0.650i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.920 - 0.389i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (209, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.920 - 0.389i)$
$L(1)$  $\approx$  $1.32601 + 0.268982i$
$L(\frac12)$  $\approx$  $1.32601 + 0.268982i$
$L(\frac{3}{2})$   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 \( 1 \)
3 \( 1 + (1.30 - 1.13i)T \)
67 \( 1 + (-1.74 - 7.99i)T \)
good5 \( 1 + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (-2.17 - 0.312i)T + (6.71 + 1.97i)T^{2} \)
11 \( 1 + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-2.69 + 1.23i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (-14.3 + 9.19i)T^{2} \)
19 \( 1 + (0.372 + 2.59i)T + (-18.2 + 5.35i)T^{2} \)
23 \( 1 + (3.27 - 22.7i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-9.91 - 4.53i)T + (20.3 + 23.4i)T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-3.26 - 11.1i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (6.68 - 46.5i)T^{2} \)
53 \( 1 + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (38.6 + 44.5i)T^{2} \)
61 \( 1 + (-3.26 - 5.08i)T + (-25.3 + 55.4i)T^{2} \)
71 \( 1 + (-59.7 - 38.3i)T^{2} \)
73 \( 1 + (-0.774 + 0.498i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-9.85 + 4.50i)T + (51.7 - 59.7i)T^{2} \)
83 \( 1 + (-34.4 - 75.4i)T^{2} \)
89 \( 1 + (12.6 + 88.0i)T^{2} \)
97 \( 1 + 19.6iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.38087731563022752781114895624, −9.634943250184338704037280649067, −8.617412384525111709091177701780, −7.895971926026955577371411017239, −6.58176593488884641425402084735, −5.90195949375533966161263024699, −4.83802919600695201299074283642, −4.21217971331135080300685429032, −2.82861639620989089966713772738, −1.03873672709058412323974492871, 1.07511815008898047306910656026, 2.23224949719835760957542690367, 3.94264313486251013822830128992, 4.94406861077144813704548522377, 5.89249941206047771508833267733, 6.62588832948977056444106016090, 7.74236432775987678710976311101, 8.185979718128660293964564320581, 9.395496855167589148268048616601, 10.43195793453150343684118244026

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