Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.987 + 0.158i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.921 + 1.07i)2-s + 3-s + (−0.300 + 1.97i)4-s + 1.54i·5-s + (0.921 + 1.07i)6-s − 4.78·7-s + (−2.39 + 1.50i)8-s + 9-s + (−1.65 + 1.42i)10-s − 3.05·11-s + (−0.300 + 1.97i)12-s − 3.50i·13-s + (−4.40 − 5.12i)14-s + 1.54i·15-s + (−3.81 − 1.18i)16-s − 4.48·17-s + ⋯
L(s)  = 1  + (0.651 + 0.758i)2-s + 0.577·3-s + (−0.150 + 0.988i)4-s + 0.691i·5-s + (0.376 + 0.437i)6-s − 1.80·7-s + (−0.847 + 0.530i)8-s + 0.333·9-s + (−0.524 + 0.451i)10-s − 0.921·11-s + (−0.0867 + 0.570i)12-s − 0.971i·13-s + (−1.17 − 1.37i)14-s + 0.399i·15-s + (−0.954 − 0.296i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.987 + 0.158i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.987 + 0.158i)$
$L(1)$  $\approx$  $0.103021 - 1.29215i$
$L(\frac12)$  $\approx$  $0.103021 - 1.29215i$
$L(\frac{3}{2})$   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 \( 1 + (-0.921 - 1.07i)T \)
3 \( 1 - T \)
67 \( 1 + (-2.49 - 7.79i)T \)
good5 \( 1 - 1.54iT - 5T^{2} \)
7 \( 1 + 4.78T + 7T^{2} \)
11 \( 1 + 3.05T + 11T^{2} \)
13 \( 1 + 3.50iT - 13T^{2} \)
17 \( 1 + 4.48T + 17T^{2} \)
19 \( 1 - 6.47iT - 19T^{2} \)
23 \( 1 - 5.90iT - 23T^{2} \)
29 \( 1 - 5.56T + 29T^{2} \)
31 \( 1 - 8.11T + 31T^{2} \)
37 \( 1 - 2.86T + 37T^{2} \)
41 \( 1 - 5.82iT - 41T^{2} \)
43 \( 1 + 8.69T + 43T^{2} \)
47 \( 1 + 3.02iT - 47T^{2} \)
53 \( 1 - 5.29iT - 53T^{2} \)
59 \( 1 - 14.0iT - 59T^{2} \)
61 \( 1 - 0.366iT - 61T^{2} \)
71 \( 1 + 4.06iT - 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + 7.10iT - 83T^{2} \)
89 \( 1 - 4.89T + 89T^{2} \)
97 \( 1 - 5.71iT - 97T^{2} \)
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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

−10.34134287934252791758320084719, −9.980216519671268919706218271585, −8.803765548583715341632631057138, −7.979564498760166638193759274737, −7.13518029186004893910879602854, −6.37568401880703966285954592119, −5.66496706709012140750028389349, −4.28253135165194985625953249611, −3.06157434537949123827083666409, −2.89448492017103826657832431130, 0.44921417443839920392097901779, 2.40356367546619797809820778357, 3.02746980146050113574303452674, 4.32158247005584204423296346431, 4.94089179648895859419591233816, 6.49797726818316097857652518747, 6.74065528009411569857732886642, 8.558050786766454005730172296177, 9.084486489382950940518554641925, 9.867018761787507025528850950375

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