Properties

Degree 2
Conductor $ 2^{4} \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 3·5-s + 2·7-s + 6·9-s + 11-s − 9·15-s − 6·17-s − 4·19-s + 6·21-s − 23-s + 4·25-s + 9·27-s − 8·29-s + 7·31-s + 3·33-s − 6·35-s − 37-s + 4·41-s − 6·43-s − 18·45-s + 8·47-s − 3·49-s − 18·51-s + 2·53-s − 3·55-s − 12·57-s + 59-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.34·5-s + 0.755·7-s + 2·9-s + 0.301·11-s − 2.32·15-s − 1.45·17-s − 0.917·19-s + 1.30·21-s − 0.208·23-s + 4/5·25-s + 1.73·27-s − 1.48·29-s + 1.25·31-s + 0.522·33-s − 1.01·35-s − 0.164·37-s + 0.624·41-s − 0.914·43-s − 2.68·45-s + 1.16·47-s − 3/7·49-s − 2.52·51-s + 0.274·53-s − 0.404·55-s − 1.58·57-s + 0.130·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(176\)    =    \(2^{4} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{176} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 176,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.655423653$
$L(\frac12)$  $\approx$  $1.655423653$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
11 \( 1 - T \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−19.82300518603389, −19.24429066123046, −18.48704761534358, −17.28544708415228, −15.92599455826993, −15.21159273944788, −14.77452103013363, −13.77326066303018, −12.89485266934258, −11.72418487570803, −10.77878158389799, −9.318682460625457, −8.441695809340898, −7.910918321149116, −6.872929086719215, −4.480901407500802, −3.743000550166358, −2.208193795973319, 2.208193795973319, 3.743000550166358, 4.480901407500802, 6.872929086719215, 7.910918321149116, 8.441695809340898, 9.318682460625457, 10.77878158389799, 11.72418487570803, 12.89485266934258, 13.77326066303018, 14.77452103013363, 15.21159273944788, 15.92599455826993, 17.28544708415228, 18.48704761534358, 19.24429066123046, 19.82300518603389

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