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-s + 3·5-s − 4·7-s − 2·9-s + 3·11-s + 5·13-s + 3·15-s − 6·17-s − 4·19-s − 4·21-s − 6·23-s + 4·25-s − 5·27-s − 29-s + 5·31-s + 3·33-s − 12·35-s + 8·37-s + 5·39-s − 43-s − 6·45-s − 3·47-s + 9·49-s − 6·51-s + 3·53-s + 9·55-s − 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s − 1.51·7-s − 2/3·9-s + 0.904·11-s + 1.38·13-s + 0.774·15-s − 1.45·17-s − 0.917·19-s − 0.872·21-s − 1.25·23-s + 4/5·25-s − 0.962·27-s − 0.185·29-s + 0.898·31-s + 0.522·33-s − 2.02·35-s + 1.31·37-s + 0.800·39-s − 0.152·43-s − 0.894·45-s − 0.437·47-s + 9/7·49-s − 0.840·51-s + 0.412·53-s + 1.21·55-s − 0.529·57-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  =  $(2,\ 116,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.276492277$
$L(\frac12)$  $\approx$  $1.276492277$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−19.99030134684646, −19.42462320770168, −18.25907605562193, −17.37666861900571, −16.49974200176219, −15.50724432716315, −14.28455177620883, −13.48368508077491, −12.99804767549760, −11.42857369641744, −10.13721881868963, −9.257030938124671, −8.557384373226300, −6.357708216143407, −6.172471311174408, −3.831799631053322, −2.337401376881860, 2.337401376881860, 3.831799631053322, 6.172471311174408, 6.357708216143407, 8.557384373226300, 9.257030938124671, 10.13721881868963, 11.42857369641744, 12.99804767549760, 13.48368508077491, 14.28455177620883, 15.50724432716315, 16.49974200176219, 17.37666861900571, 18.25907605562193, 19.42462320770168, 19.99030134684646

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