Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.992 + 0.125i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.01 + 0.985i)2-s − 3-s + (0.0573 + 1.99i)4-s − 1.66i·5-s + (−1.01 − 0.985i)6-s − 0.193·7-s + (−1.91 + 2.08i)8-s + 9-s + (1.64 − 1.69i)10-s − 5.43·11-s + (−0.0573 − 1.99i)12-s + 3.51i·13-s + (−0.196 − 0.190i)14-s + 1.66i·15-s + (−3.99 + 0.229i)16-s − 4.63·17-s + ⋯
L(s)  = 1  + (0.717 + 0.696i)2-s − 0.577·3-s + (0.0286 + 0.999i)4-s − 0.746i·5-s + (−0.414 − 0.402i)6-s − 0.0731·7-s + (−0.676 + 0.736i)8-s + 0.333·9-s + (0.520 − 0.535i)10-s − 1.63·11-s + (−0.0165 − 0.577i)12-s + 0.975i·13-s + (−0.0524 − 0.0510i)14-s + 0.430i·15-s + (−0.998 + 0.0573i)16-s − 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.992 + 0.125i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.992 + 0.125i)$
$L(1)$  $\approx$  $0.0438479 - 0.698500i$
$L(\frac12)$  $\approx$  $0.0438479 - 0.698500i$
$L(\frac{3}{2})$   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 \( 1 + (-1.01 - 0.985i)T \)
3 \( 1 + T \)
67 \( 1 + (0.790 + 8.14i)T \)
good5 \( 1 + 1.66iT - 5T^{2} \)
7 \( 1 + 0.193T + 7T^{2} \)
11 \( 1 + 5.43T + 11T^{2} \)
13 \( 1 - 3.51iT - 13T^{2} \)
17 \( 1 + 4.63T + 17T^{2} \)
19 \( 1 - 1.14iT - 19T^{2} \)
23 \( 1 - 7.82iT - 23T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 + 8.23T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 - 0.746iT - 41T^{2} \)
43 \( 1 + 3.05T + 43T^{2} \)
47 \( 1 + 2.28iT - 47T^{2} \)
53 \( 1 + 6.58iT - 53T^{2} \)
59 \( 1 - 1.92iT - 59T^{2} \)
61 \( 1 + 1.81iT - 61T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 - 5.05T + 73T^{2} \)
79 \( 1 - 8.43T + 79T^{2} \)
83 \( 1 + 8.12iT - 83T^{2} \)
89 \( 1 - 4.29T + 89T^{2} \)
97 \( 1 - 8.73iT - 97T^{2} \)
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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

−11.08528389871207457114806980311, −9.691470577048294126639475127815, −8.896743075108490047588836856072, −7.900630046556476629314575117402, −7.20441297415273002413061846845, −6.17682531675347027523421349921, −5.28481463207772592793115684477, −4.74703294400633446648541915703, −3.64713851652245961323138826768, −2.12121593870486800544132675360, 0.26991393292343422336989653963, 2.31883425424630559862041107346, 3.06152112011141986985370379020, 4.42774642304216480413988130310, 5.25472590526171154512820218876, 6.10744267930457843856575533486, 6.96005478201978937434592829044, 8.000646885969651410015421346846, 9.290961858255623400476794616957, 10.30679415821408558082472759226

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