Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.525 - 0.851i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.483 − 1.32i)2-s − 3-s + (−1.53 + 1.28i)4-s + 3.80i·5-s + (0.483 + 1.32i)6-s + 0.504·7-s + (2.44 + 1.41i)8-s + 9-s + (5.04 − 1.83i)10-s + 0.340·11-s + (1.53 − 1.28i)12-s + 4.62i·13-s + (−0.243 − 0.670i)14-s − 3.80i·15-s + (0.693 − 3.93i)16-s − 5.54·17-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s − 0.577·3-s + (−0.765 + 0.642i)4-s + 1.69i·5-s + (0.197 + 0.542i)6-s + 0.190·7-s + (0.866 + 0.499i)8-s + 0.333·9-s + (1.59 − 0.581i)10-s + 0.102·11-s + (0.442 − 0.371i)12-s + 1.28i·13-s + (−0.0651 − 0.179i)14-s − 0.981i·15-s + (0.173 − 0.984i)16-s − 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.525 - 0.851i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.525 - 0.851i)$
$L(1)$  $\approx$  $0.234345 + 0.419961i$
$L(\frac12)$  $\approx$  $0.234345 + 0.419961i$
$L(\frac{3}{2})$   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 \( 1 + (0.483 + 1.32i)T \)
3 \( 1 + T \)
67 \( 1 + (-1.18 + 8.09i)T \)
good5 \( 1 - 3.80iT - 5T^{2} \)
7 \( 1 - 0.504T + 7T^{2} \)
11 \( 1 - 0.340T + 11T^{2} \)
13 \( 1 - 4.62iT - 13T^{2} \)
17 \( 1 + 5.54T + 17T^{2} \)
19 \( 1 + 5.66iT - 19T^{2} \)
23 \( 1 - 2.26iT - 23T^{2} \)
29 \( 1 - 2.67T + 29T^{2} \)
31 \( 1 + 3.48T + 31T^{2} \)
37 \( 1 - 3.22T + 37T^{2} \)
41 \( 1 - 5.82iT - 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 4.07iT - 47T^{2} \)
53 \( 1 + 2.30iT - 53T^{2} \)
59 \( 1 + 4.68iT - 59T^{2} \)
61 \( 1 + 5.81iT - 61T^{2} \)
71 \( 1 - 9.21iT - 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 9.58T + 79T^{2} \)
83 \( 1 - 16.5iT - 83T^{2} \)
89 \( 1 + 9.37T + 89T^{2} \)
97 \( 1 + 0.449iT - 97T^{2} \)
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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

−10.83274834410683737556690120662, −9.834344660902433673947710306432, −9.212161196684189899395619655758, −8.030702436752635935592375239455, −6.85462671103991756154109059398, −6.59903312820921111035881971685, −4.94150680855447108485322559061, −4.01122663233085900983618436708, −2.86924731207534210371461120860, −1.89339493909280113130300738057, 0.29612455011257875637117248644, 1.54522114754583498000282460840, 4.03115100839493575241099695580, 4.88271161738674797694347055562, 5.51875634908461805121270604578, 6.32512048858184220355124735296, 7.53107702469340489052441627726, 8.376702805177697969212338319011, 8.817239787116236543031473828674, 9.860455667159828286453457258994

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