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  :  $\chi_{804} (785, \cdot )$
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

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

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