Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.902 + 0.429i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.487 − 1.66i)3-s + (1.18 + 1.85i)7-s + (−2.52 + 1.62i)9-s + (4.69 + 0.675i)13-s + (5.63 + 3.62i)19-s + (2.49 − 2.87i)21-s + (0.711 − 4.94i)25-s + (3.92 + 3.40i)27-s + (3.74 − 0.538i)31-s − 8.28·37-s + (−1.17 − 8.13i)39-s + (8.54 − 3.90i)43-s + (0.898 − 1.96i)49-s + (3.26 − 11.1i)57-s + (−2.04 − 1.77i)61-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)3-s + (0.449 + 0.699i)7-s + (−0.841 + 0.540i)9-s + (1.30 + 0.187i)13-s + (1.29 + 0.831i)19-s + (0.544 − 0.628i)21-s + (0.142 − 0.989i)25-s + (0.755 + 0.654i)27-s + (0.672 − 0.0966i)31-s − 1.36·37-s + (−0.187 − 1.30i)39-s + (1.30 − 0.595i)43-s + (0.128 − 0.280i)49-s + (0.433 − 1.47i)57-s + (−0.261 − 0.226i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.902 + 0.429i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (521, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.902 + 0.429i)$
$L(1)$  $\approx$  $1.49770 - 0.338359i$
$L(\frac12)$  $\approx$  $1.49770 - 0.338359i$
$L(\frac{3}{2})$   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 \( 1 + (0.487 + 1.66i)T \)
67 \( 1 + (-0.575 - 8.16i)T \)
good5 \( 1 + (-0.711 + 4.94i)T^{2} \)
7 \( 1 + (-1.18 - 1.85i)T + (-2.90 + 6.36i)T^{2} \)
11 \( 1 + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (-4.69 - 0.675i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (11.1 - 12.8i)T^{2} \)
19 \( 1 + (-5.63 - 3.62i)T + (7.89 + 17.2i)T^{2} \)
23 \( 1 + (-19.3 + 12.4i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-3.74 + 0.538i)T + (29.7 - 8.73i)T^{2} \)
37 \( 1 + 8.28T + 37T^{2} \)
41 \( 1 + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-8.54 + 3.90i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-39.5 + 25.4i)T^{2} \)
53 \( 1 + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (56.6 - 16.6i)T^{2} \)
61 \( 1 + (2.04 + 1.77i)T + (8.68 + 60.3i)T^{2} \)
71 \( 1 + (46.4 + 53.6i)T^{2} \)
73 \( 1 + (-11.1 + 12.8i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-6.35 - 0.914i)T + (75.7 + 22.2i)T^{2} \)
83 \( 1 + (11.8 - 82.1i)T^{2} \)
89 \( 1 + (-74.8 - 48.1i)T^{2} \)
97 \( 1 - 17.2iT - 97T^{2} \)
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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

−10.38206506885506148582754031930, −9.120229017077418719968560124231, −8.380360855636446041343542880953, −7.71977421270577202381416366438, −6.63727169039056468587768503544, −5.87649199794967648619012817935, −5.10998370569772686605359785889, −3.62495537327450161596182285660, −2.32081137960358165977288704353, −1.15803288825886686491589806569, 1.07110217817557382051227124621, 3.08141341726237852228146657585, 3.93904664361203673778330737775, 4.92250128659815842337596590224, 5.72581405703028424553159822375, 6.81134124784108428698329936444, 7.82847120856246495606030200424, 8.803141788642321906083188217271, 9.493248277518244853574743696907, 10.45018731751264356457327858442

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