Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.569 + 0.822i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.58 − 0.707i)3-s − 3.16·5-s + 1.41i·7-s + (2.00 − 2.23i)9-s − 3.16·11-s − 4.24i·13-s + (−5.00 + 2.23i)15-s − 6.70i·17-s − 3·19-s + (1.00 + 2.23i)21-s − 6.70i·23-s + 5.00·25-s + (1.58 − 4.94i)27-s + 2.23i·29-s + 4.24i·31-s + ⋯
L(s)  = 1  + (0.912 − 0.408i)3-s − 1.41·5-s + 0.534i·7-s + (0.666 − 0.745i)9-s − 0.953·11-s − 1.17i·13-s + (−1.29 + 0.577i)15-s − 1.62i·17-s − 0.688·19-s + (0.218 + 0.487i)21-s − 1.39i·23-s + 1.00·25-s + (0.304 − 0.952i)27-s + 0.415i·29-s + 0.762i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.569 + 0.822i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (401, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.569 + 0.822i)$
$L(1)$  $\approx$  $0.479938 - 0.915806i$
$L(\frac12)$  $\approx$  $0.479938 - 0.915806i$
$L(\frac{3}{2})$   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 \( 1 \)
3 \( 1 + (-1.58 + 0.707i)T \)
67 \( 1 + (7 - 4.24i)T \)
good5 \( 1 + 3.16T + 5T^{2} \)
7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 + 3.16T + 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 6.70iT - 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + 6.70iT - 23T^{2} \)
29 \( 1 - 2.23iT - 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + 12.6T + 41T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 - 2.23iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 2.23iT - 59T^{2} \)
61 \( 1 - 2.82iT - 61T^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 - 14.1iT - 79T^{2} \)
83 \( 1 + 4.47iT - 83T^{2} \)
89 \( 1 + 2.23iT - 89T^{2} \)
97 \( 1 + 8.48iT - 97T^{2} \)
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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

−9.978351469892347463343713549106, −8.616878376809080674107488731152, −8.435100683883083931434271068388, −7.46985906217230104649218205236, −6.89916270789847921638707219963, −5.39442805229894832653822403570, −4.38563512973051706941597439941, −3.19376664674168012437709767066, −2.54219591982894105706580875697, −0.44631356795939100241948177450, 1.89843080662219645565068105752, 3.39686480807023915682682180802, 4.03755883383147708236504169715, 4.77324740785394044542911710850, 6.36067843233360608287864755165, 7.56457466248675989732278274294, 7.908024280113900994066628508722, 8.693406440329866169819453156618, 9.691594017124287663749422174347, 10.56230427035530359621520271541

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