Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 7^{2} $
Sign $0.968 - 0.250i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.999·8-s + 4·13-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)17-s + (1 − 1.73i)19-s + (2.5 + 4.33i)25-s + (−2 + 3.46i)26-s + 6·29-s + (−2 − 3.46i)31-s + (−0.499 − 0.866i)32-s + 6·34-s + (−1 + 1.73i)37-s + (0.999 + 1.73i)38-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.353·8-s + 1.10·13-s + (−0.125 + 0.216i)16-s + (−0.727 − 1.26i)17-s + (0.229 − 0.397i)19-s + (0.5 + 0.866i)25-s + (−0.392 + 0.679i)26-s + 1.11·29-s + (−0.359 − 0.622i)31-s + (−0.0883 − 0.153i)32-s + 1.02·34-s + (−0.164 + 0.284i)37-s + (0.162 + 0.280i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $0.968 - 0.250i$
motivic weight  =  \(1\)
character  :  $\chi_{882} (667, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 882,\ (\ :1/2),\ 0.968 - 0.250i)$
$L(1)$  $\approx$  $1.28668 + 0.163967i$
$L(\frac12)$  $\approx$  $1.28668 + 0.163967i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.03458217196504737449000820073, −9.077121502901718740174244911167, −8.657303558006121919107339150908, −7.50832315445137166254741236384, −6.86895803606696537734860737541, −5.91763966675191839057885172732, −5.01874547052489253303943963197, −3.96201716102204789433231923865, −2.58645387986878904012478155109, −0.905061614027266334376927730819, 1.13998151026283401648955805326, 2.44197013065682108198827134917, 3.67099487377760487735754976729, 4.47193653024812030918892525950, 5.83842875049177632595080084384, 6.63343478235961428792921448827, 7.82916649232246112174780567500, 8.584248611012648524612602635586, 9.174995102540854900453790019561, 10.39247262359629443990942940583

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