Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.947 + 0.319i$
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 + (4.93 − 1.19i)7-s + (2.87 + 0.845i)9-s + (−3.38 + 1.16i)13-s + (−0.539 + 2.22i)19-s + (−8.75 + 0.835i)21-s + (3.27 − 3.77i)25-s + (−4.72 − 2.15i)27-s + (8.77 + 3.03i)31-s + (5.89 − 10.2i)37-s + (6.08 − 1.17i)39-s + (4.69 + 7.31i)43-s + (16.6 − 8.60i)49-s + (1.47 − 3.68i)57-s + (1.00 − 0.712i)61-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)3-s + (1.86 − 0.452i)7-s + (0.959 + 0.281i)9-s + (−0.937 + 0.324i)13-s + (−0.123 + 0.510i)19-s + (−1.91 + 0.182i)21-s + (0.654 − 0.755i)25-s + (−0.909 − 0.415i)27-s + (1.57 + 0.545i)31-s + (0.968 − 1.67i)37-s + (0.974 − 0.187i)39-s + (0.716 + 1.11i)43-s + (2.38 − 1.22i)49-s + (0.195 − 0.487i)57-s + (0.128 − 0.0912i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.947 + 0.319i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (221, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.947 + 0.319i)$
$L(1)$  $\approx$  $1.34666 - 0.220835i$
$L(\frac12)$  $\approx$  $1.34666 - 0.220835i$
$L(\frac{3}{2})$   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 \( 1 + (-8.18 + 0.186i)T \)
good5 \( 1 + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (-4.93 + 1.19i)T + (6.22 - 3.20i)T^{2} \)
11 \( 1 + (-3.59 - 10.3i)T^{2} \)
13 \( 1 + (3.38 - 1.16i)T + (10.2 - 8.03i)T^{2} \)
17 \( 1 + (16.9 - 1.61i)T^{2} \)
19 \( 1 + (0.539 - 2.22i)T + (-16.8 - 8.70i)T^{2} \)
23 \( 1 + (-5.42 - 22.3i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.77 - 3.03i)T + (24.3 + 19.1i)T^{2} \)
37 \( 1 + (-5.89 + 10.2i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (23.7 + 33.3i)T^{2} \)
43 \( 1 + (-4.69 - 7.31i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + (-34.0 + 32.4i)T^{2} \)
53 \( 1 + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (8.39 - 58.3i)T^{2} \)
61 \( 1 + (-1.00 + 0.712i)T + (19.9 - 57.6i)T^{2} \)
71 \( 1 + (70.6 + 6.74i)T^{2} \)
73 \( 1 + (6.53 + 9.18i)T + (-23.8 + 68.9i)T^{2} \)
79 \( 1 + (-0.356 + 1.85i)T + (-73.3 - 29.3i)T^{2} \)
83 \( 1 + (-81.5 - 15.7i)T^{2} \)
89 \( 1 + (85.3 - 25.0i)T^{2} \)
97 \( 1 + (1.93 + 1.11i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.53499788203272365856295800094, −9.534898090131101328704401955315, −8.249733510813530159912822027109, −7.64591786900580765729860735369, −6.78603488612048224043697019921, −5.69247185615307465227142054418, −4.73617072071392353403210923627, −4.31341382714870503495557330541, −2.24362751798995147354041188690, −1.02981508869601206801733379309, 1.14482444600786693838432202829, 2.51067961521154217638890476000, 4.37002661852931099887251967995, 4.95120724621188905979045449559, 5.65908144990363488869885984029, 6.83824084129934489640138206493, 7.72566200797362554676053935283, 8.504923816965440192590178367553, 9.606125487429205304770719160023, 10.47056336168064350586222607187

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