Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.895 + 0.444i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.921 − 1.07i)2-s − 3-s + (−0.300 + 1.97i)4-s − 1.54i·5-s + (0.921 + 1.07i)6-s + 4.78·7-s + (2.39 − 1.50i)8-s + 9-s + (−1.65 + 1.42i)10-s + 3.05·11-s + (0.300 − 1.97i)12-s + 3.50i·13-s + (−4.40 − 5.12i)14-s + 1.54i·15-s + (−3.81 − 1.18i)16-s − 4.48·17-s + ⋯
L(s)  = 1  + (−0.651 − 0.758i)2-s − 0.577·3-s + (−0.150 + 0.988i)4-s − 0.691i·5-s + (0.376 + 0.437i)6-s + 1.80·7-s + (0.847 − 0.530i)8-s + 0.333·9-s + (−0.524 + 0.451i)10-s + 0.921·11-s + (0.0867 − 0.570i)12-s + 0.971i·13-s + (−1.17 − 1.37i)14-s + 0.399i·15-s + (−0.954 − 0.296i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.895 + 0.444i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.895 + 0.444i)$
$L(1)$  $\approx$  $1.10799 - 0.259808i$
$L(\frac12)$  $\approx$  $1.10799 - 0.259808i$
$L(\frac{3}{2})$   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 \( 1 + (0.921 + 1.07i)T \)
3 \( 1 + T \)
67 \( 1 + (2.49 - 7.79i)T \)
good5 \( 1 + 1.54iT - 5T^{2} \)
7 \( 1 - 4.78T + 7T^{2} \)
11 \( 1 - 3.05T + 11T^{2} \)
13 \( 1 - 3.50iT - 13T^{2} \)
17 \( 1 + 4.48T + 17T^{2} \)
19 \( 1 - 6.47iT - 19T^{2} \)
23 \( 1 - 5.90iT - 23T^{2} \)
29 \( 1 - 5.56T + 29T^{2} \)
31 \( 1 + 8.11T + 31T^{2} \)
37 \( 1 - 2.86T + 37T^{2} \)
41 \( 1 + 5.82iT - 41T^{2} \)
43 \( 1 - 8.69T + 43T^{2} \)
47 \( 1 + 3.02iT - 47T^{2} \)
53 \( 1 + 5.29iT - 53T^{2} \)
59 \( 1 - 14.0iT - 59T^{2} \)
61 \( 1 + 0.366iT - 61T^{2} \)
71 \( 1 + 4.06iT - 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + 7.10iT - 83T^{2} \)
89 \( 1 - 4.89T + 89T^{2} \)
97 \( 1 + 5.71iT - 97T^{2} \)
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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

−10.40312930701617745142006453351, −9.141877158808022802439272864738, −8.800375507689634439477835772823, −7.80958448548773575734072767737, −7.00328198971430988055426772366, −5.60912724789033010935073872921, −4.51239678732247572593855925567, −4.00761837950874640602689548268, −1.91190040165403171099378037626, −1.28721531746563933478051822696, 0.942458577335231169673621104163, 2.36415472493996013899398591312, 4.47361131911997999855175954646, 4.98395230673110149703507953144, 6.17317580032660480961324654279, 6.88775427334829457209888409865, 7.68735720796807022216730138863, 8.567175713681758645678410854423, 9.262937693822999194518174337490, 10.62651742082049398720953552246

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