Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.305 - 0.952i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.956 + 1.04i)2-s + 3-s + (−0.168 + 1.99i)4-s − 1.36i·5-s + (0.956 + 1.04i)6-s + 1.56·7-s + (−2.23 + 1.73i)8-s + 9-s + (1.41 − 1.30i)10-s + 0.528·11-s + (−0.168 + 1.99i)12-s + 2.52i·13-s + (1.49 + 1.63i)14-s − 1.36i·15-s + (−3.94 − 0.671i)16-s + 5.97·17-s + ⋯
L(s)  = 1  + (0.676 + 0.736i)2-s + 0.577·3-s + (−0.0841 + 0.996i)4-s − 0.609i·5-s + (0.390 + 0.425i)6-s + 0.592·7-s + (−0.790 + 0.612i)8-s + 0.333·9-s + (0.448 − 0.412i)10-s + 0.159·11-s + (−0.0485 + 0.575i)12-s + 0.699i·13-s + (0.400 + 0.435i)14-s − 0.351i·15-s + (−0.985 − 0.167i)16-s + 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.305 - 0.952i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.305 - 0.952i)$
$L(1)$  $\approx$  $2.27978 + 1.66194i$
$L(\frac12)$  $\approx$  $2.27978 + 1.66194i$
$L(\frac{3}{2})$   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 \( 1 + (-0.956 - 1.04i)T \)
3 \( 1 - T \)
67 \( 1 + (7.97 + 1.83i)T \)
good5 \( 1 + 1.36iT - 5T^{2} \)
7 \( 1 - 1.56T + 7T^{2} \)
11 \( 1 - 0.528T + 11T^{2} \)
13 \( 1 - 2.52iT - 13T^{2} \)
17 \( 1 - 5.97T + 17T^{2} \)
19 \( 1 - 2.51iT - 19T^{2} \)
23 \( 1 - 2.87iT - 23T^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 - 8.15T + 31T^{2} \)
37 \( 1 + 7.80T + 37T^{2} \)
41 \( 1 + 11.2iT - 41T^{2} \)
43 \( 1 - 3.82T + 43T^{2} \)
47 \( 1 - 3.73iT - 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 - 8.44iT - 59T^{2} \)
61 \( 1 - 2.04iT - 61T^{2} \)
71 \( 1 + 4.62iT - 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 2.66T + 79T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 - 2.96T + 89T^{2} \)
97 \( 1 + 13.7iT - 97T^{2} \)
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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

−10.31977795344550378985975445236, −9.228265766943792270229613219771, −8.604942534821345443805076624677, −7.77469858830957380364382343932, −7.13411602410049051261718000366, −5.88964647733137575902280175502, −5.09126081491252256776909650256, −4.15190984189346451393057826882, −3.24197726647804782595241139320, −1.69729307490600841819676987197, 1.28516962955804143452422108945, 2.67981943288559532368331824804, 3.35750533775337788420410056871, 4.52583461260720264710067595390, 5.44717331744737904505648151337, 6.49794671671763508667445669485, 7.52563296511294564135428971055, 8.445666742567241292804996120638, 9.493462434252992616910019033933, 10.25900660930030713564380573949

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