Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.121 - 0.992i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.57 + 0.719i)3-s + (−3.36 − 2.91i)7-s + (1.96 − 2.26i)9-s + (1.63 + 2.54i)13-s + (4.23 + 4.89i)19-s + (7.39 + 2.17i)21-s + (−4.20 + 2.70i)25-s + (−1.46 + 4.98i)27-s + (−6.00 + 9.33i)31-s + 12.1·37-s + (−4.40 − 2.83i)39-s + (−12.9 − 1.85i)43-s + (1.82 + 12.6i)49-s + (−10.2 − 4.65i)57-s + (−4.39 + 14.9i)61-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)3-s + (−1.27 − 1.10i)7-s + (0.654 − 0.755i)9-s + (0.453 + 0.705i)13-s + (0.972 + 1.12i)19-s + (1.61 + 0.474i)21-s + (−0.841 + 0.540i)25-s + (−0.281 + 0.959i)27-s + (−1.07 + 1.67i)31-s + 1.99·37-s + (−0.705 − 0.453i)39-s + (−1.97 − 0.283i)43-s + (0.260 + 1.81i)49-s + (−1.35 − 0.617i)57-s + (−0.563 + 1.91i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.121 - 0.992i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.121 - 0.992i)$
$L(1)$  $\approx$  $0.431835 + 0.488089i$
$L(\frac12)$  $\approx$  $0.431835 + 0.488089i$
$L(\frac{3}{2})$   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 \( 1 + (1.57 - 0.719i)T \)
67 \( 1 + (-5.79 + 5.78i)T \)
good5 \( 1 + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (3.36 + 2.91i)T + (0.996 + 6.92i)T^{2} \)
11 \( 1 + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-1.63 - 2.54i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (-4.23 - 4.89i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (15.0 - 17.3i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (6.00 - 9.33i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 - 12.1T + 37T^{2} \)
41 \( 1 + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (12.9 + 1.85i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (30.7 - 35.5i)T^{2} \)
53 \( 1 + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (4.39 - 14.9i)T + (-51.3 - 32.9i)T^{2} \)
71 \( 1 + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (-5.46 - 1.60i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (-9.04 - 14.0i)T + (-32.8 + 71.8i)T^{2} \)
83 \( 1 + (-69.8 + 44.8i)T^{2} \)
89 \( 1 + (58.2 + 67.2i)T^{2} \)
97 \( 1 - 6.97iT - 97T^{2} \)
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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

−10.37807484587014335816469859187, −9.823855832931944090465214437108, −9.146361670114100373876052145747, −7.68982946860865905980184811376, −6.84514953456152490999074587030, −6.19700225336367005655039930765, −5.21659051716985217165840584439, −3.96398172182685627390699410753, −3.44762618334476231942376883039, −1.24276202379162434812750458360, 0.40515711029627455510024832309, 2.27093230583196490948465701826, 3.41193702598138738284360745320, 4.87872879191395068358972557385, 5.87918702234907039733908890693, 6.24993565735497314702906765617, 7.33010946127707678816350449370, 8.242123284461205948597781820560, 9.484682569489210891643452231736, 9.841221091492245139526657776135

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