Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.934 - 0.355i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.31 + 0.523i)2-s + 3-s + (1.45 + 1.37i)4-s − 1.71i·5-s + (1.31 + 0.523i)6-s + 3.43·7-s + (1.18 + 2.56i)8-s + 9-s + (0.898 − 2.25i)10-s − 1.17·11-s + (1.45 + 1.37i)12-s − 1.11i·13-s + (4.51 + 1.79i)14-s − 1.71i·15-s + (0.216 + 3.99i)16-s − 6.00·17-s + ⋯
L(s)  = 1  + (0.928 + 0.370i)2-s + 0.577·3-s + (0.725 + 0.687i)4-s − 0.767i·5-s + (0.536 + 0.213i)6-s + 1.29·7-s + (0.419 + 0.907i)8-s + 0.333·9-s + (0.284 − 0.713i)10-s − 0.352·11-s + (0.419 + 0.397i)12-s − 0.308i·13-s + (1.20 + 0.480i)14-s − 0.443i·15-s + (0.0540 + 0.998i)16-s − 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.934 - 0.355i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.934 - 0.355i)$
$L(1)$  $\approx$  $3.51043 + 0.645083i$
$L(\frac12)$  $\approx$  $3.51043 + 0.645083i$
$L(\frac{3}{2})$   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 \( 1 + (-1.31 - 0.523i)T \)
3 \( 1 - T \)
67 \( 1 + (-3.55 + 7.37i)T \)
good5 \( 1 + 1.71iT - 5T^{2} \)
7 \( 1 - 3.43T + 7T^{2} \)
11 \( 1 + 1.17T + 11T^{2} \)
13 \( 1 + 1.11iT - 13T^{2} \)
17 \( 1 + 6.00T + 17T^{2} \)
19 \( 1 - 1.61iT - 19T^{2} \)
23 \( 1 + 5.58iT - 23T^{2} \)
29 \( 1 + 2.67T + 29T^{2} \)
31 \( 1 + 3.52T + 31T^{2} \)
37 \( 1 - 4.31T + 37T^{2} \)
41 \( 1 - 8.77iT - 41T^{2} \)
43 \( 1 + 1.55T + 43T^{2} \)
47 \( 1 + 3.12iT - 47T^{2} \)
53 \( 1 - 9.63iT - 53T^{2} \)
59 \( 1 - 5.48iT - 59T^{2} \)
61 \( 1 - 3.17iT - 61T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + 7.90T + 73T^{2} \)
79 \( 1 + 7.48T + 79T^{2} \)
83 \( 1 - 3.57iT - 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 - 7.37iT - 97T^{2} \)
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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

−10.59863885175621285067990488971, −9.118043461176080644169187137626, −8.371106136698250587419518556909, −7.86207109776515127246710132688, −6.85003759020925367904581990739, −5.70100299660100248977211937460, −4.69264067490950624413426755136, −4.31401342263972377953317655174, −2.79516538981435118062498645560, −1.71074193425939038851993381880, 1.75860376628843748484150101993, 2.58528977520734837209782284806, 3.77135555251272621168646754516, 4.66956942315491959141093798845, 5.57234324196385284767361165253, 6.83685438903207551941382026324, 7.39022762134554158194013120681, 8.482938219095441468166826305497, 9.480702251339453875900143202712, 10.51611218132981782465530262722

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