Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.577 + 0.816i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (−1.41 + i)3-s − 2.00·4-s + 1.41i·5-s + (1.41 + 2.00i)6-s − 2i·7-s + 2.82i·8-s + (1.00 − 2.82i)9-s + 2.00·10-s − 5.65·11-s + (2.82 − 2.00i)12-s + 2·13-s − 2.82·14-s + (−1.41 − 2.00i)15-s + 4.00·16-s + 5.65i·17-s + ⋯
L(s)  = 1  − 0.999i·2-s + (−0.816 + 0.577i)3-s − 1.00·4-s + 0.632i·5-s + (0.577 + 0.816i)6-s − 0.755i·7-s + 1.00i·8-s + (0.333 − 0.942i)9-s + 0.632·10-s − 1.70·11-s + (0.816 − 0.577i)12-s + 0.554·13-s − 0.755·14-s + (−0.365 − 0.516i)15-s + 1.00·16-s + 1.37i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.577 + 0.816i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (671, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.577 + 0.816i)$
$L(1)$  $\approx$  $0.799048 - 0.413617i$
$L(\frac12)$  $\approx$  $0.799048 - 0.413617i$
$L(\frac{3}{2})$   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 \( 1 + 1.41iT \)
3 \( 1 + (1.41 - i)T \)
67 \( 1 + iT \)
good5 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 5.65T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 7.07T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 7.07T + 47T^{2} \)
53 \( 1 + 12.7iT - 53T^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 8iT - 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 - 16.9iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−10.41033235415812928324281282632, −9.741384678566851761138298750787, −8.640876723454655335709521679784, −7.63963407630896693890836180260, −6.49911245637268527186132071557, −5.43175626954294552499861907331, −4.59070768999280769743554230802, −3.63680614981927736974151880905, −2.60670422944462861464638233188, −0.74290849436228213523786050826, 0.892092873694361216457947448404, 2.79696675267356136779345135149, 4.65753866281560205616606545555, 5.31584971960212456468378711703, 5.80437583900437488492381753774, 6.99028103061584082549561968000, 7.67396443625619550237195295669, 8.515338425947997247766127515727, 9.269690580028461862601677820556, 10.38826209531835086402998694210

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