Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.227 + 0.973i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.936 − 1.45i)3-s + (4.81 − 2.19i)7-s + (−1.24 − 2.72i)9-s + (−0.298 − 1.01i)13-s + (−2.01 + 4.41i)19-s + (1.30 − 9.07i)21-s + (4.79 − 1.40i)25-s + (−5.14 − 0.739i)27-s + (−1.53 + 5.21i)31-s − 7.64·37-s + (−1.75 − 0.516i)39-s + (5.83 − 5.05i)43-s + (13.7 − 15.8i)49-s + (4.54 + 7.07i)57-s + (1.72 + 0.247i)61-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)3-s + (1.81 − 0.830i)7-s + (−0.415 − 0.909i)9-s + (−0.0827 − 0.281i)13-s + (−0.462 + 1.01i)19-s + (0.284 − 1.97i)21-s + (0.959 − 0.281i)25-s + (−0.989 − 0.142i)27-s + (−0.275 + 0.937i)31-s − 1.25·37-s + (−0.281 − 0.0827i)39-s + (0.889 − 0.770i)43-s + (1.96 − 2.26i)49-s + (0.602 + 0.937i)57-s + (0.220 + 0.0317i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.227 + 0.973i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (53, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.227 + 0.973i)$
$L(1)$  $\approx$  $1.68721 - 1.33844i$
$L(\frac12)$  $\approx$  $1.68721 - 1.33844i$
$L(\frac{3}{2})$   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 \( 1 + (-0.936 + 1.45i)T \)
67 \( 1 + (8.16 - 0.591i)T \)
good5 \( 1 + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (-4.81 + 2.19i)T + (4.58 - 5.29i)T^{2} \)
11 \( 1 + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (0.298 + 1.01i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (2.41 - 16.8i)T^{2} \)
19 \( 1 + (2.01 - 4.41i)T + (-12.4 - 14.3i)T^{2} \)
23 \( 1 + (-9.55 - 20.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1.53 - 5.21i)T + (-26.0 - 16.7i)T^{2} \)
37 \( 1 + 7.64T + 37T^{2} \)
41 \( 1 + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-5.83 + 5.05i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 + (-19.5 - 42.7i)T^{2} \)
53 \( 1 + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-1.72 - 0.247i)T + (58.5 + 17.1i)T^{2} \)
71 \( 1 + (10.1 + 70.2i)T^{2} \)
73 \( 1 + (1.74 - 12.1i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (4.36 + 14.8i)T + (-66.4 + 42.7i)T^{2} \)
83 \( 1 + (79.6 - 23.3i)T^{2} \)
89 \( 1 + (-36.9 + 80.9i)T^{2} \)
97 \( 1 - 19.2iT - 97T^{2} \)
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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

−10.29560170040771338049410067470, −8.835292324167611368022974427288, −8.330618005004032445965012720305, −7.54713372206303708353934693704, −6.92603743868490162681160215886, −5.65449769196373938542769834015, −4.62488888230319745489050312619, −3.55739008509824932105025603298, −2.09156115006110979873075139247, −1.14443104812082336923551578578, 1.84287196062897168595852298711, 2.82165545247707761898046785724, 4.30825724939015439339850304991, 4.89964204310630340610801581701, 5.73379683375644523216148262053, 7.23170829517441776652969949160, 8.143325429791524527839422861502, 8.785374038007723895245519693619, 9.353354032301703426160622877985, 10.59859486470338050288666630682

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