Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.872 - 0.488i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.351i)2-s + 3-s + (1.75 − 0.962i)4-s + 3.58i·5-s + (−1.36 + 0.351i)6-s − 4.07·7-s + (−2.06 + 1.93i)8-s + 9-s + (−1.25 − 4.91i)10-s + 4.30·11-s + (1.75 − 0.962i)12-s + 0.588i·13-s + (5.58 − 1.43i)14-s + 3.58i·15-s + (2.14 − 3.37i)16-s + 0.330·17-s + ⋯
L(s)  = 1  + (−0.968 + 0.248i)2-s + 0.577·3-s + (0.876 − 0.481i)4-s + 1.60i·5-s + (−0.559 + 0.143i)6-s − 1.53·7-s + (−0.729 + 0.683i)8-s + 0.333·9-s + (−0.398 − 1.55i)10-s + 1.29·11-s + (0.506 − 0.277i)12-s + 0.163i·13-s + (1.49 − 0.382i)14-s + 0.926i·15-s + (0.537 − 0.843i)16-s + 0.0801·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.872 - 0.488i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.872 - 0.488i)$
$L(1)$  $\approx$  $0.195437 + 0.749310i$
$L(\frac12)$  $\approx$  $0.195437 + 0.749310i$
$L(\frac{3}{2})$   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 \( 1 + (1.36 - 0.351i)T \)
3 \( 1 - T \)
67 \( 1 + (4.33 + 6.94i)T \)
good5 \( 1 - 3.58iT - 5T^{2} \)
7 \( 1 + 4.07T + 7T^{2} \)
11 \( 1 - 4.30T + 11T^{2} \)
13 \( 1 - 0.588iT - 13T^{2} \)
17 \( 1 - 0.330T + 17T^{2} \)
19 \( 1 - 5.25iT - 19T^{2} \)
23 \( 1 + 0.501iT - 23T^{2} \)
29 \( 1 + 5.91T + 29T^{2} \)
31 \( 1 + 5.74T + 31T^{2} \)
37 \( 1 + 1.03T + 37T^{2} \)
41 \( 1 - 7.04iT - 41T^{2} \)
43 \( 1 + 6.86T + 43T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 + 8.47iT - 53T^{2} \)
59 \( 1 + 8.93iT - 59T^{2} \)
61 \( 1 - 6.57iT - 61T^{2} \)
71 \( 1 - 0.0930iT - 71T^{2} \)
73 \( 1 + 6.32T + 73T^{2} \)
79 \( 1 - 1.43T + 79T^{2} \)
83 \( 1 - 12.5iT - 83T^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
97 \( 1 + 6.83iT - 97T^{2} \)
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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

−10.31945289617809225369078789620, −9.647160634388343186014663916985, −9.216255081350013013532170611410, −7.981879848362902708838644529640, −7.10686589377452951093690280998, −6.53660089149370024583088864604, −5.97450808618861997240659565349, −3.65733354603696824069249596438, −3.14725486165401049980010020647, −1.87189712500231537846592132828, 0.47578252746269515868587240733, 1.79284190505599544569310697677, 3.26871729873623146449436332415, 4.10071351153499934688235671310, 5.61240468946522131430626246389, 6.70143494022582928099337829734, 7.44363992581001306591371127363, 8.720031821058947428782479040504, 9.072407205583867273241532509930, 9.481495516611157552722764599130

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