Properties

Degree 2
Conductor $ 2^{2} \cdot 29 $
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 + 4·7-s + 6·9-s − 11-s − 3·13-s − 9·15-s + 2·17-s + 4·19-s − 12·21-s − 6·23-s + 4·25-s − 9·27-s − 29-s + 9·31-s + 3·33-s + 12·35-s − 8·37-s + 9·39-s − 8·41-s − 5·43-s + 18·45-s − 7·47-s + 9·49-s − 6·51-s − 5·53-s − 3·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.34·5-s + 1.51·7-s + 2·9-s − 0.301·11-s − 0.832·13-s − 2.32·15-s + 0.485·17-s + 0.917·19-s − 2.61·21-s − 1.25·23-s + 4/5·25-s − 1.73·27-s − 0.185·29-s + 1.61·31-s + 0.522·33-s + 2.02·35-s − 1.31·37-s + 1.44·39-s − 1.24·41-s − 0.762·43-s + 2.68·45-s − 1.02·47-s + 9/7·49-s − 0.840·51-s − 0.686·53-s − 0.404·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(116\)    =    \(2^{2} \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{116} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 116,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8589770376$
$L(\frac12)$  $\approx$  $0.8589770376$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
29 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−18.69170374323725, −17.86280579619829, −17.50712566564851, −16.85890580011111, −15.72059469938346, −14.39708646727799, −13.54107010958861, −12.15935897292602, −11.60485318971593, −10.41077210374199, −9.835259972878506, −7.950296694455790, −6.562825125594148, −5.398093774132946, −4.910462455043474, −1.678614401636443, 1.678614401636443, 4.910462455043474, 5.398093774132946, 6.562825125594148, 7.950296694455790, 9.835259972878506, 10.41077210374199, 11.60485318971593, 12.15935897292602, 13.54107010958861, 14.39708646727799, 15.72059469938346, 16.85890580011111, 17.50712566564851, 17.86280579619829, 18.69170374323725

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