Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.860 + 0.509i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.41i)3-s − 2·5-s + (−0.949 − 0.548i)7-s + (−1.00 − 2.82i)9-s + (0.724 − 1.25i)11-s + (4.5 − 2.59i)13-s + (−2 + 2.82i)15-s + (−5.17 + 2.98i)17-s + (−1.5 − 2.59i)19-s + (−1.72 + 0.794i)21-s + (−0.825 + 0.476i)23-s − 25-s + (−5.00 − 1.41i)27-s + (−4.62 − 2.66i)29-s + (−1.5 − 0.866i)31-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)3-s − 0.894·5-s + (−0.358 − 0.207i)7-s + (−0.333 − 0.942i)9-s + (0.218 − 0.378i)11-s + (1.24 − 0.720i)13-s + (−0.516 + 0.730i)15-s + (−1.25 + 0.724i)17-s + (−0.344 − 0.596i)19-s + (−0.376 + 0.173i)21-s + (−0.172 + 0.0994i)23-s − 0.200·25-s + (−0.962 − 0.272i)27-s + (−0.858 − 0.495i)29-s + (−0.269 − 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.860 + 0.509i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (365, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.860 + 0.509i)$
$L(1)$  $\approx$  $0.273049 - 0.995930i$
$L(\frac12)$  $\approx$  $0.273049 - 0.995930i$
$L(\frac{3}{2})$   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 \( 1 \)
3 \( 1 + (-1 + 1.41i)T \)
67 \( 1 + (-8 - 1.73i)T \)
good5 \( 1 + 2T + 5T^{2} \)
7 \( 1 + (0.949 + 0.548i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.724 + 1.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.17 - 2.98i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.825 - 0.476i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.62 + 2.66i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.94 + 8.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.72 - 2.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 1.27iT - 43T^{2} \)
47 \( 1 + (0.275 + 0.158i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 0.635iT - 59T^{2} \)
61 \( 1 + (-9.39 + 5.42i)T + (30.5 - 52.8i)T^{2} \)
71 \( 1 + (-11.1 - 6.45i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.94 - 5.10i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.398 - 0.230i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.0 - 5.81i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.73iT - 89T^{2} \)
97 \( 1 + (-11.8 + 6.84i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.791931121141788016153070242392, −8.623494072389611927595989610770, −8.395379907763502545382756816553, −7.34791836989052170388315673152, −6.58383774324913178166195040662, −5.72313604307986687771539784995, −3.98759498332930193505150608106, −3.51374106422397149560284641196, −2.08165071876625340917275967084, −0.46308863549140384750307188062, 2.05437118195364086643866712957, 3.48618449586255284097732342985, 4.04683176555877781936001640893, 5.01761970851814346606948405147, 6.32364909425578328219476656322, 7.25410243528187087338243911908, 8.328498121704622611852105630872, 8.876240615322327879161021488910, 9.628230741226306898101211077296, 10.64170164139124281138222471733

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