Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.364 - 0.931i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 + 0.418i)2-s + 3-s + (1.64 + 1.13i)4-s + 1.44i·5-s + (1.35 + 0.418i)6-s − 2.69·7-s + (1.75 + 2.21i)8-s + 9-s + (−0.606 + 1.95i)10-s + 3.96·11-s + (1.64 + 1.13i)12-s + 2.95i·13-s + (−3.64 − 1.13i)14-s + 1.44i·15-s + (1.44 + 3.73i)16-s − 2.16·17-s + ⋯
L(s)  = 1  + (0.955 + 0.296i)2-s + 0.577·3-s + (0.824 + 0.565i)4-s + 0.647i·5-s + (0.551 + 0.170i)6-s − 1.01·7-s + (0.620 + 0.784i)8-s + 0.333·9-s + (−0.191 + 0.618i)10-s + 1.19·11-s + (0.476 + 0.326i)12-s + 0.818i·13-s + (−0.974 − 0.302i)14-s + 0.374i·15-s + (0.360 + 0.932i)16-s − 0.524·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.364 - 0.931i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.364 - 0.931i)$
$L(1)$  $\approx$  $2.65221 + 1.80985i$
$L(\frac12)$  $\approx$  $2.65221 + 1.80985i$
$L(\frac{3}{2})$   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 \( 1 + (-1.35 - 0.418i)T \)
3 \( 1 - T \)
67 \( 1 + (1.85 + 7.97i)T \)
good5 \( 1 - 1.44iT - 5T^{2} \)
7 \( 1 + 2.69T + 7T^{2} \)
11 \( 1 - 3.96T + 11T^{2} \)
13 \( 1 - 2.95iT - 13T^{2} \)
17 \( 1 + 2.16T + 17T^{2} \)
19 \( 1 + 6.52iT - 19T^{2} \)
23 \( 1 - 5.27iT - 23T^{2} \)
29 \( 1 - 1.48T + 29T^{2} \)
31 \( 1 + 4.41T + 31T^{2} \)
37 \( 1 + 6.67T + 37T^{2} \)
41 \( 1 - 0.555iT - 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + 3.34iT - 53T^{2} \)
59 \( 1 + 7.97iT - 59T^{2} \)
61 \( 1 - 0.843iT - 61T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + 8.46T + 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 + 8.13iT - 83T^{2} \)
89 \( 1 - 3.50T + 89T^{2} \)
97 \( 1 - 8.55iT - 97T^{2} \)
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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

−10.59163399386018502860478758247, −9.326339046824135470430657902662, −8.907928684104956130260968873428, −7.43096615874866402837123060170, −6.78068283119757461134462206489, −6.35516166286290243304087527801, −4.94240956059300168136337780331, −3.83138448385229060235607081992, −3.19707252018805760246493770712, −2.04800330262655361597712788800, 1.24023237265147984001456552647, 2.68361773736208323754420254863, 3.68350614047523364509051350886, 4.40039828046411802646869554749, 5.67464345035867085976265911320, 6.42858711078666648619807142077, 7.32847941561764995296948822341, 8.530719351240092430123800545572, 9.322655950205960555294248862725, 10.15587416083614081121412303531

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