Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.585 + 0.810i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 0.0966i)2-s + 3-s + (1.98 − 0.272i)4-s − 2.83i·5-s + (1.41 − 0.0966i)6-s − 0.830·7-s + (2.76 − 0.576i)8-s + 9-s + (−0.274 − 4.00i)10-s − 3.46·11-s + (1.98 − 0.272i)12-s − 0.104i·13-s + (−1.17 + 0.0802i)14-s − 2.83i·15-s + (3.85 − 1.08i)16-s + 4.98·17-s + ⋯
L(s)  = 1  + (0.997 − 0.0683i)2-s + 0.577·3-s + (0.990 − 0.136i)4-s − 1.27i·5-s + (0.576 − 0.0394i)6-s − 0.313·7-s + (0.979 − 0.203i)8-s + 0.333·9-s + (−0.0867 − 1.26i)10-s − 1.04·11-s + (0.571 − 0.0786i)12-s − 0.0289i·13-s + (−0.313 + 0.0214i)14-s − 0.733i·15-s + (0.962 − 0.270i)16-s + 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.585 + 0.810i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.585 + 0.810i)$
$L(1)$  $\approx$  $3.00625 - 1.53781i$
$L(\frac12)$  $\approx$  $3.00625 - 1.53781i$
$L(\frac{3}{2})$   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 \( 1 + (-1.41 + 0.0966i)T \)
3 \( 1 - T \)
67 \( 1 + (-3.84 - 7.22i)T \)
good5 \( 1 + 2.83iT - 5T^{2} \)
7 \( 1 + 0.830T + 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 0.104iT - 13T^{2} \)
17 \( 1 - 4.98T + 17T^{2} \)
19 \( 1 + 5.61iT - 19T^{2} \)
23 \( 1 - 7.10iT - 23T^{2} \)
29 \( 1 + 0.746T + 29T^{2} \)
31 \( 1 - 2.40T + 31T^{2} \)
37 \( 1 - 0.364T + 37T^{2} \)
41 \( 1 - 9.73iT - 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 - 3.75iT - 47T^{2} \)
53 \( 1 + 6.44iT - 53T^{2} \)
59 \( 1 - 14.0iT - 59T^{2} \)
61 \( 1 + 8.51iT - 61T^{2} \)
71 \( 1 + 6.72iT - 71T^{2} \)
73 \( 1 - 14.6T + 73T^{2} \)
79 \( 1 - 2.89T + 79T^{2} \)
83 \( 1 - 12.2iT - 83T^{2} \)
89 \( 1 - 0.183T + 89T^{2} \)
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

−10.01928765950891058962690787310, −9.422815015014758818449728834495, −8.210870254991386877608618585235, −7.66940252948585179928501251749, −6.52931953562635824185698569183, −5.25821490392579886250942241033, −4.93730119752406066197380545322, −3.64456989802182290831100622819, −2.72701260928392865148347242356, −1.29245347244703587821544456822, 2.12036808894156588090047681403, 3.06499965936229710707308081138, 3.67486135382804195717784665064, 5.02321699798468100888156391315, 6.05024093559703994288658230477, 6.83101457528017886155731686222, 7.67026102608788564857275181633, 8.335624851321378086870569000685, 10.09721090737108354495189322049, 10.25485886903075419746622860314

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