Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.555 - 0.831i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.487 − 1.66i)3-s + (−5.07 − 0.241i)7-s + (−2.52 − 1.62i)9-s + (−0.619 + 1.54i)13-s + (0.411 + 8.64i)19-s + (−2.87 + 8.31i)21-s + (0.711 + 4.94i)25-s + (−3.92 + 3.40i)27-s + (−3.51 − 8.77i)31-s + (−2.71 + 4.70i)37-s + (2.27 + 1.78i)39-s + (−11.9 − 5.44i)43-s + (18.6 + 1.78i)49-s + (14.5 + 3.53i)57-s + (−2.88 − 14.9i)61-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)3-s + (−1.91 − 0.0913i)7-s + (−0.841 − 0.540i)9-s + (−0.171 + 0.429i)13-s + (0.0944 + 1.98i)19-s + (−0.627 + 1.81i)21-s + (0.142 + 0.989i)25-s + (−0.755 + 0.654i)27-s + (−0.630 − 1.57i)31-s + (−0.446 + 0.773i)37-s + (0.363 + 0.285i)39-s + (−1.81 − 0.830i)43-s + (2.67 + 0.255i)49-s + (1.92 + 0.467i)57-s + (−0.369 − 1.91i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.555 - 0.831i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (677, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.555 - 0.831i)$
$L(1)$  $\approx$  $0.0577913 + 0.108135i$
$L(\frac12)$  $\approx$  $0.0577913 + 0.108135i$
$L(\frac{3}{2})$   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 \( 1 + (-0.487 + 1.66i)T \)
67 \( 1 + (5.21 - 6.30i)T \)
good5 \( 1 + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (5.07 + 0.241i)T + (6.96 + 0.665i)T^{2} \)
11 \( 1 + (10.2 + 4.08i)T^{2} \)
13 \( 1 + (0.619 - 1.54i)T + (-9.40 - 8.97i)T^{2} \)
17 \( 1 + (5.56 - 16.0i)T^{2} \)
19 \( 1 + (-0.411 - 8.64i)T + (-18.9 + 1.80i)T^{2} \)
23 \( 1 + (-1.09 + 22.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.51 + 8.77i)T + (-22.4 + 21.3i)T^{2} \)
37 \( 1 + (2.71 - 4.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (40.2 - 7.75i)T^{2} \)
43 \( 1 + (11.9 + 5.44i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + (41.7 - 21.5i)T^{2} \)
53 \( 1 + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (56.6 + 16.6i)T^{2} \)
61 \( 1 + (2.88 + 14.9i)T + (-56.6 + 22.6i)T^{2} \)
71 \( 1 + (23.2 + 67.0i)T^{2} \)
73 \( 1 + (16.0 - 3.09i)T + (67.7 - 27.1i)T^{2} \)
79 \( 1 + (9.02 + 11.4i)T + (-18.6 + 76.7i)T^{2} \)
83 \( 1 + (65.2 - 51.3i)T^{2} \)
89 \( 1 + (-74.8 + 48.1i)T^{2} \)
97 \( 1 + (-17.0 - 9.84i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.32335426371478302557910466655, −9.654473203721449531709054718479, −8.911689812146901310745235467886, −7.85779829698315127261535105635, −7.04498104827752154666569045451, −6.30928556716684059665431317962, −5.62599298115897733896846579683, −3.78414866512268940030846745815, −3.11999632433440524111083149091, −1.75381957194294533166424924937, 0.05486594066442224109001702813, 2.75160225619651011796247963920, 3.26017498078397684232374547075, 4.45384137638100508706306858533, 5.45559901419639676806118314186, 6.47509515750856448787068986624, 7.25476731215024649335300366054, 8.717355271885687708308051874969, 9.099857495109772317056515600524, 10.04341258822457820491770932447

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