Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 2·7-s − 2·9-s + 11-s − 4·13-s + 3·15-s + 6·17-s − 8·19-s + 2·21-s + 3·23-s + 4·25-s + 5·27-s − 5·31-s − 33-s + 6·35-s − 37-s + 4·39-s + 10·43-s + 6·45-s − 3·49-s − 6·51-s − 6·53-s − 3·55-s + 8·57-s − 3·59-s − 4·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.755·7-s − 2/3·9-s + 0.301·11-s − 1.10·13-s + 0.774·15-s + 1.45·17-s − 1.83·19-s + 0.436·21-s + 0.625·23-s + 4/5·25-s + 0.962·27-s − 0.898·31-s − 0.174·33-s + 1.01·35-s − 0.164·37-s + 0.640·39-s + 1.52·43-s + 0.894·45-s − 3/7·49-s − 0.840·51-s − 0.824·53-s − 0.404·55-s + 1.05·57-s − 0.390·59-s − 0.512·61-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  =  1
Selberg data  =  $(2,\ 176,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
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

−19.97450285740642, −19.30322937306692, −18.91408415110407, −17.39124385780718, −16.78754290502430, −16.08940433933179, −14.97592597972922, −14.40747608810486, −12.70519142939999, −12.25900555383232, −11.33372769095584, −10.42805632650209, −9.146174464181464, −7.992911069953808, −6.993468367991627, −5.787196353912736, −4.404003716941649, −3.101393177977803, 0, 3.101393177977803, 4.404003716941649, 5.787196353912736, 6.993468367991627, 7.992911069953808, 9.146174464181464, 10.42805632650209, 11.33372769095584, 12.25900555383232, 12.70519142939999, 14.40747608810486, 14.97592597972922, 16.08940433933179, 16.78754290502430, 17.39124385780718, 18.91408415110407, 19.30322937306692, 19.97450285740642

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