Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.988 - 0.148i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.530 − 1.31i)2-s + 3-s + (−1.43 + 1.39i)4-s + 2.85i·5-s + (−0.530 − 1.31i)6-s + 4.91·7-s + (2.58 + 1.14i)8-s + 9-s + (3.74 − 1.51i)10-s − 6.48·11-s + (−1.43 + 1.39i)12-s − 2.89i·13-s + (−2.60 − 6.44i)14-s + 2.85i·15-s + (0.131 − 3.99i)16-s + 4.14·17-s + ⋯
L(s)  = 1  + (−0.375 − 0.927i)2-s + 0.577·3-s + (−0.718 + 0.695i)4-s + 1.27i·5-s + (−0.216 − 0.535i)6-s + 1.85·7-s + (0.914 + 0.405i)8-s + 0.333·9-s + (1.18 − 0.478i)10-s − 1.95·11-s + (−0.414 + 0.401i)12-s − 0.802i·13-s + (−0.696 − 1.72i)14-s + 0.737i·15-s + (0.0329 − 0.999i)16-s + 1.00·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.988 - 0.148i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.988 - 0.148i)$
$L(1)$  $\approx$  $1.62342 + 0.121620i$
$L(\frac12)$  $\approx$  $1.62342 + 0.121620i$
$L(\frac{3}{2})$   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 \( 1 + (0.530 + 1.31i)T \)
3 \( 1 - T \)
67 \( 1 + (6.66 + 4.75i)T \)
good5 \( 1 - 2.85iT - 5T^{2} \)
7 \( 1 - 4.91T + 7T^{2} \)
11 \( 1 + 6.48T + 11T^{2} \)
13 \( 1 + 2.89iT - 13T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 - 4.77iT - 19T^{2} \)
23 \( 1 - 6.27iT - 23T^{2} \)
29 \( 1 - 4.52T + 29T^{2} \)
31 \( 1 - 0.791T + 31T^{2} \)
37 \( 1 - 2.50T + 37T^{2} \)
41 \( 1 - 7.22iT - 41T^{2} \)
43 \( 1 + 4.34T + 43T^{2} \)
47 \( 1 - 1.23iT - 47T^{2} \)
53 \( 1 + 0.123iT - 53T^{2} \)
59 \( 1 - 2.67iT - 59T^{2} \)
61 \( 1 + 4.75iT - 61T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + 1.84T + 73T^{2} \)
79 \( 1 - 0.728T + 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 - 0.704T + 89T^{2} \)
97 \( 1 + 9.89iT - 97T^{2} \)
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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

−10.42258062249661463256036714186, −9.803008224833294853268863838409, −8.283614100324291927510604746564, −7.87516335408369703443398101831, −7.51757087857360222626817645391, −5.58376274378716017451170661230, −4.76264165543374006312978331360, −3.36089127958555195095682140941, −2.68378796080770454969270083076, −1.57599902424493073151238140548, 0.967106419799678623066359907242, 2.27998593890034866991910230538, 4.48979964268628046552613219341, 4.85522906045869145221338669236, 5.56651950845221702365276046085, 7.16053176229597243037020917548, 7.921651035739545084955885352987, 8.463268409529841783016441470225, 8.888609673112493284466080796405, 10.09717533282652050464056621098

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