Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.677 + 0.735i$
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 + (−4.40 − 0.633i)7-s + (0.426 − 2.96i)9-s + (0.903 − 0.412i)13-s + (−1.11 − 7.77i)19-s + (−6.48 + 4.16i)21-s + (−2.07 − 4.54i)25-s + (−2.80 − 4.37i)27-s + (−5.98 − 2.73i)31-s − 2.14·37-s + (0.714 − 1.56i)39-s + (0.833 + 2.84i)43-s + (12.2 + 3.60i)49-s + (−10.2 − 8.91i)57-s + (8.17 + 12.7i)61-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)3-s + (−1.66 − 0.239i)7-s + (0.142 − 0.989i)9-s + (0.250 − 0.114i)13-s + (−0.256 − 1.78i)19-s + (−1.41 + 0.909i)21-s + (−0.415 − 0.909i)25-s + (−0.540 − 0.841i)27-s + (−1.07 − 0.490i)31-s − 0.353·37-s + (0.114 − 0.250i)39-s + (0.127 + 0.433i)43-s + (1.75 + 0.515i)49-s + (−1.36 − 1.18i)57-s + (1.04 + 1.62i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.677 + 0.735i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (209, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.677 + 0.735i)$
$L(1)$  $\approx$  $0.485295 - 1.10641i$
$L(\frac12)$  $\approx$  $0.485295 - 1.10641i$
$L(\frac{3}{2})$   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 \( 1 + (-4.89 - 6.55i)T \)
good5 \( 1 + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (4.40 + 0.633i)T + (6.71 + 1.97i)T^{2} \)
11 \( 1 + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-0.903 + 0.412i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (-14.3 + 9.19i)T^{2} \)
19 \( 1 + (1.11 + 7.77i)T + (-18.2 + 5.35i)T^{2} \)
23 \( 1 + (3.27 - 22.7i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (5.98 + 2.73i)T + (20.3 + 23.4i)T^{2} \)
37 \( 1 + 2.14T + 37T^{2} \)
41 \( 1 + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-0.833 - 2.84i)T + (-36.1 + 23.2i)T^{2} \)
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 \( 1 + (-8.17 - 12.7i)T + (-25.3 + 55.4i)T^{2} \)
71 \( 1 + (-59.7 - 38.3i)T^{2} \)
73 \( 1 + (-13.3 + 8.59i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (3.23 - 1.47i)T + (51.7 - 59.7i)T^{2} \)
83 \( 1 + (-34.4 - 75.4i)T^{2} \)
89 \( 1 + (12.6 + 88.0i)T^{2} \)
97 \( 1 + 1.50iT - 97T^{2} \)
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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

−9.688356055813852162076417046472, −9.168711903688756443372447075008, −8.309159685728270883520084964374, −7.16558522211637344563947919618, −6.70432215280314195926781783975, −5.80182922908346268573034938718, −4.19348656007546199267295418587, −3.23873013760806586352863718128, −2.37035919796877405161449492500, −0.51882023754894552378988458545, 2.05344342786902451312619556157, 3.44065418741744724454369923448, 3.73840566239127110652581072731, 5.27795214761012019444861746853, 6.16489465445187062114239965893, 7.18086272808988978365852399320, 8.182103209597858954818855063375, 9.051689395406096657361484512449, 9.709145077776597231288619978211, 10.26265252719185090946042589639

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