Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.168 - 0.985i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 − 1.13i)3-s + (−1.35 − 1.72i)7-s + (0.426 + 2.96i)9-s + (−3.42 + 2.44i)13-s + (−2.06 − 1.62i)19-s + (−0.181 + 3.80i)21-s + (−2.07 + 4.54i)25-s + (2.80 − 4.37i)27-s + (6.03 + 4.29i)31-s + (−5.22 + 9.05i)37-s + (7.25 + 0.693i)39-s + (−3.12 + 10.6i)43-s + (0.512 − 2.11i)49-s + (0.859 + 4.45i)57-s + (4.33 + 8.40i)61-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)3-s + (−0.513 − 0.652i)7-s + (0.142 + 0.989i)9-s + (−0.951 + 0.677i)13-s + (−0.472 − 0.371i)19-s + (−0.0395 + 0.829i)21-s + (−0.415 + 0.909i)25-s + (0.540 − 0.841i)27-s + (1.08 + 0.771i)31-s + (−0.859 + 1.48i)37-s + (1.16 + 0.110i)39-s + (−0.477 + 1.62i)43-s + (0.0732 − 0.301i)49-s + (0.113 + 0.590i)57-s + (0.554 + 1.07i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.168 - 0.985i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (749, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.168 - 0.985i)$
$L(1)$  $\approx$  $0.249326 + 0.295703i$
$L(\frac12)$  $\approx$  $0.249326 + 0.295703i$
$L(\frac{3}{2})$   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 \( 1 + (1.30 + 1.13i)T \)
67 \( 1 + (7.79 - 2.48i)T \)
good5 \( 1 + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (1.35 + 1.72i)T + (-1.65 + 6.80i)T^{2} \)
11 \( 1 + (6.38 + 8.96i)T^{2} \)
13 \( 1 + (3.42 - 2.44i)T + (4.25 - 12.2i)T^{2} \)
17 \( 1 + (-0.808 + 16.9i)T^{2} \)
19 \( 1 + (2.06 + 1.62i)T + (4.47 + 18.4i)T^{2} \)
23 \( 1 + (18.0 - 14.2i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.03 - 4.29i)T + (10.1 + 29.2i)T^{2} \)
37 \( 1 + (5.22 - 9.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-36.4 + 18.7i)T^{2} \)
43 \( 1 + (3.12 - 10.6i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 + (-43.6 - 17.4i)T^{2} \)
53 \( 1 + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (-4.33 - 8.40i)T + (-35.3 + 49.6i)T^{2} \)
71 \( 1 + (-3.37 - 70.9i)T^{2} \)
73 \( 1 + (12.7 - 6.56i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (0.645 + 6.75i)T + (-77.5 + 14.9i)T^{2} \)
83 \( 1 + (82.6 - 7.88i)T^{2} \)
89 \( 1 + (12.6 - 88.0i)T^{2} \)
97 \( 1 + (-7.53 - 4.34i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.41249934462290529413311328631, −9.914460570727191390517781482003, −8.754131587327077690016208744459, −7.68218166218803243025744060515, −6.92356564931902207611108783121, −6.36969435122869956573122165119, −5.15962769184190815331566685912, −4.34363473323951292142027957609, −2.86482293334213423208500389516, −1.44222247659244239236892694565, 0.21247503377628491252959026537, 2.39771650576121480330341376894, 3.63889130236300847618501255584, 4.70140629049627106818495775733, 5.63004573117448117958548466930, 6.28021918955062062973048010414, 7.34492866459729062641945549297, 8.481067288232725708856856576541, 9.368325581677782959056787732475, 10.12171084864216562084197653910

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