Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.981 + 0.192i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 − 0.159i)2-s − 3-s + (1.94 − 0.449i)4-s + 0.947i·5-s + (−1.40 + 0.159i)6-s + 0.0251·7-s + (2.66 − 0.943i)8-s + 9-s + (0.151 + 1.33i)10-s + 1.77·11-s + (−1.94 + 0.449i)12-s − 5.89i·13-s + (0.0352 − 0.00401i)14-s − 0.947i·15-s + (3.59 − 1.75i)16-s + 2.08·17-s + ⋯
L(s)  = 1  + (0.993 − 0.113i)2-s − 0.577·3-s + (0.974 − 0.224i)4-s + 0.423i·5-s + (−0.573 + 0.0652i)6-s + 0.00949·7-s + (0.942 − 0.333i)8-s + 0.333·9-s + (0.0478 + 0.420i)10-s + 0.536·11-s + (−0.562 + 0.129i)12-s − 1.63i·13-s + (0.00943 − 0.00107i)14-s − 0.244i·15-s + (0.899 − 0.437i)16-s + 0.505·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.981 + 0.192i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.981 + 0.192i)$
$L(1)$  $\approx$  $2.60425 - 0.252683i$
$L(\frac12)$  $\approx$  $2.60425 - 0.252683i$
$L(\frac{3}{2})$   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 \( 1 + (-1.40 + 0.159i)T \)
3 \( 1 + T \)
67 \( 1 + (7.47 + 3.33i)T \)
good5 \( 1 - 0.947iT - 5T^{2} \)
7 \( 1 - 0.0251T + 7T^{2} \)
11 \( 1 - 1.77T + 11T^{2} \)
13 \( 1 + 5.89iT - 13T^{2} \)
17 \( 1 - 2.08T + 17T^{2} \)
19 \( 1 - 4.96iT - 19T^{2} \)
23 \( 1 - 4.05iT - 23T^{2} \)
29 \( 1 - 5.86T + 29T^{2} \)
31 \( 1 + 3.69T + 31T^{2} \)
37 \( 1 - 5.82T + 37T^{2} \)
41 \( 1 - 6.60iT - 41T^{2} \)
43 \( 1 + 2.25T + 43T^{2} \)
47 \( 1 + 11.4iT - 47T^{2} \)
53 \( 1 + 7.44iT - 53T^{2} \)
59 \( 1 + 2.53iT - 59T^{2} \)
61 \( 1 - 2.42iT - 61T^{2} \)
71 \( 1 - 1.82iT - 71T^{2} \)
73 \( 1 + 9.08T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 - 14.6iT - 83T^{2} \)
89 \( 1 + 9.99T + 89T^{2} \)
97 \( 1 + 3.04iT - 97T^{2} \)
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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

−10.31406606143309409560048834653, −9.920388152094385006758211684802, −8.257027687803053515271548110460, −7.43536280356336710007780834137, −6.49134175933622020574158202758, −5.73091452907411997576561901389, −4.99952570517827333238417473764, −3.74648737514014434572456629692, −2.94184356685440328740494465036, −1.31411386668856942105846531027, 1.41209145249051837638126203132, 2.84224690903354188106104920676, 4.35431545117535832440125418937, 4.64790499742034403569990416625, 5.87915382832811950144521413886, 6.65671290627054659829716250761, 7.29298586905142790506246119801, 8.606009589870327638563051267354, 9.433633049182183304830339447061, 10.61602614663685506554397246386

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