Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.299 + 0.954i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.57 − 0.719i)3-s + (−2.61 − 2.26i)7-s + (1.96 − 2.26i)9-s + (−3.27 − 5.08i)13-s + (0.112 + 0.130i)19-s + (−5.76 − 1.69i)21-s + (−4.20 + 2.70i)25-s + (1.46 − 4.98i)27-s + (4.78 − 7.44i)31-s + 4.66·37-s + (−8.81 − 5.66i)39-s + (4.37 + 0.628i)43-s + (0.713 + 4.96i)49-s + (0.271 + 0.123i)57-s + (−2.77 + 9.45i)61-s + ⋯
L(s)  = 1  + (0.909 − 0.415i)3-s + (−0.990 − 0.857i)7-s + (0.654 − 0.755i)9-s + (−0.906 − 1.41i)13-s + (0.0258 + 0.0298i)19-s + (−1.25 − 0.369i)21-s + (−0.841 + 0.540i)25-s + (0.281 − 0.959i)27-s + (0.858 − 1.33i)31-s + 0.767·37-s + (−1.41 − 0.906i)39-s + (0.666 + 0.0958i)43-s + (0.101 + 0.709i)49-s + (0.0359 + 0.0164i)57-s + (−0.355 + 1.21i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.299 + 0.954i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.299 + 0.954i)$
$L(1)$  $\approx$  $0.941614 - 1.28265i$
$L(\frac12)$  $\approx$  $0.941614 - 1.28265i$
$L(\frac{3}{2})$   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 \( 1 + (-1.57 + 0.719i)T \)
67 \( 1 + (-7.66 + 2.86i)T \)
good5 \( 1 + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (2.61 + 2.26i)T + (0.996 + 6.92i)T^{2} \)
11 \( 1 + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (3.27 + 5.08i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (-0.112 - 0.130i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (15.0 - 17.3i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-4.78 + 7.44i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 - 4.66T + 37T^{2} \)
41 \( 1 + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-4.37 - 0.628i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (30.7 - 35.5i)T^{2} \)
53 \( 1 + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (2.77 - 9.45i)T + (-51.3 - 32.9i)T^{2} \)
71 \( 1 + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (-12.9 - 3.80i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (6.70 + 10.4i)T + (-32.8 + 71.8i)T^{2} \)
83 \( 1 + (-69.8 + 44.8i)T^{2} \)
89 \( 1 + (58.2 + 67.2i)T^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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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

−9.844681158513820624723628540923, −9.357606651164545172522612561698, −8.034932238292826804403628062261, −7.60202453608228821044105776368, −6.71813396476164118873547369548, −5.73203969448517276979015690873, −4.28525145580636895424251866743, −3.35300488740003404118464047510, −2.46157314571744770846193522579, −0.68795928121257683908574283989, 2.08565717034992582689485617635, 2.92942554270687265989024366746, 4.05856746381306746296568665773, 4.99769719329902560033195689133, 6.26569756529640753149033209454, 7.07393348369750492701855749528, 8.130935341294797067363968619569, 9.008580576238362019075452375741, 9.564380228311552607090123358244, 10.13838453633499029167485995815

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