Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.966 - 0.256i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.71 + 0.246i)3-s + (0.211 − 0.721i)7-s + (2.87 + 0.845i)9-s + (4.59 + 3.98i)13-s + (−4.38 + 1.28i)19-s + (0.540 − 1.18i)21-s + (3.27 − 3.77i)25-s + (4.72 + 2.15i)27-s + (6.01 − 5.20i)31-s − 1.31·37-s + (6.90 + 7.96i)39-s + (0.410 + 0.638i)43-s + (5.41 + 3.47i)49-s + (−7.83 + 1.12i)57-s + (−13.5 − 6.16i)61-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)3-s + (0.0800 − 0.272i)7-s + (0.959 + 0.281i)9-s + (1.27 + 1.10i)13-s + (−1.00 + 0.295i)19-s + (0.118 − 0.258i)21-s + (0.654 − 0.755i)25-s + (0.909 + 0.415i)27-s + (1.07 − 0.935i)31-s − 0.215·37-s + (1.10 + 1.27i)39-s + (0.0626 + 0.0974i)43-s + (0.773 + 0.497i)49-s + (−1.03 + 0.149i)57-s + (−1.72 − 0.789i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.966 - 0.256i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (137, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.966 - 0.256i)$
$L(1)$  $\approx$  $2.29988 + 0.300436i$
$L(\frac12)$  $\approx$  $2.29988 + 0.300436i$
$L(\frac{3}{2})$   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 \( 1 + (-1.71 - 0.246i)T \)
67 \( 1 + (6.54 - 4.91i)T \)
good5 \( 1 + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (-0.211 + 0.721i)T + (-5.88 - 3.78i)T^{2} \)
11 \( 1 + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-4.59 - 3.98i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (-7.06 + 15.4i)T^{2} \)
19 \( 1 + (4.38 - 1.28i)T + (15.9 - 10.2i)T^{2} \)
23 \( 1 + (22.0 + 6.47i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-6.01 + 5.20i)T + (4.41 - 30.6i)T^{2} \)
37 \( 1 + 1.31T + 37T^{2} \)
41 \( 1 + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-0.410 - 0.638i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + (45.0 + 13.2i)T^{2} \)
53 \( 1 + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (8.39 - 58.3i)T^{2} \)
61 \( 1 + (13.5 + 6.16i)T + (39.9 + 46.1i)T^{2} \)
71 \( 1 + (-29.4 - 64.5i)T^{2} \)
73 \( 1 + (3.51 - 7.68i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (6.28 + 5.44i)T + (11.2 + 78.1i)T^{2} \)
83 \( 1 + (54.3 - 62.7i)T^{2} \)
89 \( 1 + (85.3 - 25.0i)T^{2} \)
97 \( 1 - 11.8iT - 97T^{2} \)
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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

−10.27693548039535402793097963399, −9.278505750488498204389650813214, −8.622618151682489628018152457857, −7.947638824565995532831774685241, −6.86234914234425077553803893356, −6.10568557408944063890546125504, −4.49399317453609243010018186887, −3.97132519750827666013877479622, −2.70957272439770908089193550130, −1.49541170821458975316981055400, 1.31401557785857073433834958468, 2.71650178409919559283455103800, 3.56469762772794240905966577406, 4.69891896333880917384515797858, 5.92332551844499486206168225323, 6.84821685722107753700422456094, 7.85548313781108221536026473771, 8.606203699726745535456469872505, 9.072148162091989284578193919036, 10.31414415169145739463148836545

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