Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.609 + 0.792i$
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 + (0.686 + 0.313i)7-s + (−1.24 + 2.72i)9-s + (1.32 − 4.50i)13-s + (−3.57 − 7.82i)19-s + (−0.185 − 1.29i)21-s + (4.79 + 1.40i)25-s + (5.14 − 0.739i)27-s + (−0.698 − 2.37i)31-s − 11.5·37-s + (−7.80 + 2.29i)39-s + (−9.23 − 8.00i)43-s + (−4.21 − 4.86i)49-s + (−8.05 + 12.5i)57-s + (13.2 − 1.90i)61-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)3-s + (0.259 + 0.118i)7-s + (−0.415 + 0.909i)9-s + (0.367 − 1.25i)13-s + (−0.819 − 1.79i)19-s + (−0.0405 − 0.282i)21-s + (0.959 + 0.281i)25-s + (0.989 − 0.142i)27-s + (−0.125 − 0.427i)31-s − 1.89·37-s + (−1.25 + 0.367i)39-s + (−1.40 − 1.22i)43-s + (−0.601 − 0.694i)49-s + (−1.06 + 1.66i)57-s + (1.69 − 0.244i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.609 + 0.792i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (713, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.609 + 0.792i)$
$L(1)$  $\approx$  $0.430625 - 0.874834i$
$L(\frac12)$  $\approx$  $0.430625 - 0.874834i$
$L(\frac{3}{2})$   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 \( 1 + (7.18 + 3.91i)T \)
good5 \( 1 + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.686 - 0.313i)T + (4.58 + 5.29i)T^{2} \)
11 \( 1 + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-1.32 + 4.50i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (2.41 + 16.8i)T^{2} \)
19 \( 1 + (3.57 + 7.82i)T + (-12.4 + 14.3i)T^{2} \)
23 \( 1 + (-9.55 + 20.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (0.698 + 2.37i)T + (-26.0 + 16.7i)T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (9.23 + 8.00i)T + (6.11 + 42.5i)T^{2} \)
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 \( 1 + (-13.2 + 1.90i)T + (58.5 - 17.1i)T^{2} \)
71 \( 1 + (10.1 - 70.2i)T^{2} \)
73 \( 1 + (-2.15 - 14.9i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-4.99 + 17.0i)T + (-66.4 - 42.7i)T^{2} \)
83 \( 1 + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (-36.9 - 80.9i)T^{2} \)
97 \( 1 - 4.08iT - 97T^{2} \)
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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

−10.23805190802194476232765062306, −8.803718711215544336986250585609, −8.313980488391664220224142552191, −7.19200143941364873635099610861, −6.62023466263329346705610667476, −5.47997721503969984397763124713, −4.86399647997287368104665638917, −3.24163032098975185337678331793, −2.03983713517344529464595399512, −0.52449979567729259925374496248, 1.64031235169965337597291560820, 3.40206171787609144113767829355, 4.26255876158784994377756764460, 5.11790575367471966500193673441, 6.18833168909016534732373230577, 6.85678960969084207486037384313, 8.255302302096136298549601700207, 8.891666395601446122853560587484, 9.886088045166193116377739543204, 10.52004413115373393660138584523

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