Properties

Degree 2
Conductor $ 2^{2} \cdot 13 $
Sign $0.711 + 0.702i$
Motivic weight 0
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 − 5-s + 0.999·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 0.999·18-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)26-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)32-s − 0.999·34-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 5-s + 0.999·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 0.999·18-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)26-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)32-s − 0.999·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(52\)    =    \(2^{2} \cdot 13\)
\( \varepsilon \)  =  $0.711 + 0.702i$
motivic weight  =  \(0\)
character  :  $\chi_{52} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 52,\ (\ :0),\ 0.711 + 0.702i)$
$L(\frac{1}{2})$  $\approx$  $0.3248332447$
$L(\frac12)$  $\approx$  $0.3248332447$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;13\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−15.87294873793952215742661356483, −14.36669503053971576318634699798, −13.06765961975395203124255805669, −11.93591325352333454299037842375, −11.08546064930377023602876475562, −9.896784789924800205671186547144, −8.339948637340426939401113737946, −7.53136053749764317001096925301, −4.87537373101459305382914912093, −3.05212385545771777947032732532, 4.16168360291039984927090654039, 6.04668941269644592254494882943, 7.39766540537995423654738286953, 8.532649398844231199448686039577, 9.709118972498971638443287007627, 11.24632239324280198745131969122, 12.40865268975567652680377933239, 14.12767115902838684115562210190, 14.96124058887919042375486728325, 15.86901024324633578237109774276

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