Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.989 - 0.143i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.338 + 1.37i)2-s + 3-s + (−1.77 − 0.929i)4-s − 0.0609i·5-s + (−0.338 + 1.37i)6-s − 3.75·7-s + (1.87 − 2.11i)8-s + 9-s + (0.0837 + 0.0206i)10-s + 1.10·11-s + (−1.77 − 0.929i)12-s − 4.93i·13-s + (1.27 − 5.15i)14-s − 0.0609i·15-s + (2.27 + 3.29i)16-s + 6.72·17-s + ⋯
L(s)  = 1  + (−0.239 + 0.970i)2-s + 0.577·3-s + (−0.885 − 0.464i)4-s − 0.0272i·5-s + (−0.138 + 0.560i)6-s − 1.42·7-s + (0.662 − 0.748i)8-s + 0.333·9-s + (0.0264 + 0.00652i)10-s + 0.332·11-s + (−0.511 − 0.268i)12-s − 1.37i·13-s + (0.339 − 1.37i)14-s − 0.0157i·15-s + (0.568 + 0.822i)16-s + 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.989 - 0.143i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.989 - 0.143i)$
$L(1)$  $\approx$  $1.28875 + 0.0928688i$
$L(\frac12)$  $\approx$  $1.28875 + 0.0928688i$
$L(\frac{3}{2})$   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 \( 1 + (0.338 - 1.37i)T \)
3 \( 1 - T \)
67 \( 1 + (6.62 - 4.80i)T \)
good5 \( 1 + 0.0609iT - 5T^{2} \)
7 \( 1 + 3.75T + 7T^{2} \)
11 \( 1 - 1.10T + 11T^{2} \)
13 \( 1 + 4.93iT - 13T^{2} \)
17 \( 1 - 6.72T + 17T^{2} \)
19 \( 1 + 1.13iT - 19T^{2} \)
23 \( 1 + 2.63iT - 23T^{2} \)
29 \( 1 + 1.58T + 29T^{2} \)
31 \( 1 - 0.470T + 31T^{2} \)
37 \( 1 - 4.49T + 37T^{2} \)
41 \( 1 + 1.53iT - 41T^{2} \)
43 \( 1 - 3.78T + 43T^{2} \)
47 \( 1 + 8.21iT - 47T^{2} \)
53 \( 1 + 2.81iT - 53T^{2} \)
59 \( 1 + 8.36iT - 59T^{2} \)
61 \( 1 - 4.04iT - 61T^{2} \)
71 \( 1 + 8.01iT - 71T^{2} \)
73 \( 1 - 11.0T + 73T^{2} \)
79 \( 1 + 4.16T + 79T^{2} \)
83 \( 1 - 0.0220iT - 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 - 3.14iT - 97T^{2} \)
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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

−9.977314396002437616595633272167, −9.388323138660779397651661341575, −8.495355382242394568635511406353, −7.71138320889461690797469570620, −6.88697667275195398119878946256, −6.03124081931059122059441959144, −5.17108433576593701277941208596, −3.77384611061001282465502379588, −2.97154821736184309146323301152, −0.76369307194305425382900134419, 1.30987792082583937168570111222, 2.75585350739350622338908144707, 3.51173616399850174605962825542, 4.38770916975481680777208307957, 5.82095907939965038827970964900, 6.94394921669330893319930727183, 7.83606903156291285014707235835, 8.947096735753170706052316472097, 9.507562467587276206277398632801, 9.981952599349691809167750008747

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