Properties

Degree 2
Conductor $ 2^{3} \cdot 11^{2} $
Sign $0.426 + 0.904i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 1.41i·7-s − 15-s − 1.41i·17-s + 1.41i·21-s − 23-s + 27-s + 31-s − 1.41i·35-s + 37-s − 1.41i·43-s − 1.00·49-s + 1.41i·51-s + 59-s + ⋯
L(s)  = 1  − 3-s + 5-s − 1.41i·7-s − 15-s − 1.41i·17-s + 1.41i·21-s − 23-s + 27-s + 31-s − 1.41i·35-s + 37-s − 1.41i·43-s − 1.00·49-s + 1.41i·51-s + 59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(968\)    =    \(2^{3} \cdot 11^{2}\)
\( \varepsilon \)  =  $0.426 + 0.904i$
motivic weight  =  \(0\)
character  :  $\chi_{968} (241, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 968,\ (\ :0),\ 0.426 + 0.904i)$
$L(\frac{1}{2})$  $\approx$  $0.7886763449$
$L(\frac12)$  $\approx$  $0.7886763449$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + T + T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.04854837437942496760166223865, −9.647044327939144600882331096177, −8.368606414729696413023233107418, −7.29596142280420772880598090374, −6.59390574852220492798975328233, −5.76192932390643374241656158249, −4.97351158894988733359554889810, −3.97475509500580425100787307385, −2.49730313996834084159477406008, −0.888355604703299204175892473574, 1.76567963821866631400230613918, 2.80834934473539215359122660250, 4.42751356559661842221418965394, 5.55201673953473326287727927844, 5.96257813594915694726073057534, 6.46429044551159361631351075448, 8.043006352600327529359166118573, 8.731718541648375726483639070846, 9.698767793115177574493064887325, 10.29899130671281585511177224199

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