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  + 2·3-s − 2·5-s + 4·7-s + 9-s − 6·11-s + 2·13-s − 4·15-s + 2·17-s − 6·19-s + 8·21-s + 4·23-s − 25-s − 4·27-s − 29-s − 6·31-s − 12·33-s − 8·35-s + 2·37-s + 4·39-s + 2·41-s + 10·43-s − 2·45-s − 2·47-s + 9·49-s + 4·51-s + 10·53-s + 12·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 1.03·15-s + 0.485·17-s − 1.37·19-s + 1.74·21-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 0.185·29-s − 1.07·31-s − 2.08·33-s − 1.35·35-s + 0.328·37-s + 0.640·39-s + 0.312·41-s + 1.52·43-s − 0.298·45-s − 0.291·47-s + 9/7·49-s + 0.560·51-s + 1.37·53-s + 1.61·55-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  =  $(2,\ 116,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.323119098$
$L(\frac12)$  $\approx$  $1.323119098$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.20853250133328, −18.51427961106496, −17.54613534341227, −16.20205000493621, −15.12474620287817, −14.79172056037881, −13.64913618416886, −12.70905976606632, −11.29976427066418, −10.62296992663263, −8.881475557679452, −8.042165076629360, −7.613462457402513, −5.332743547825179, −3.964897420015663, −2.388482371431609, 2.388482371431609, 3.964897420015663, 5.332743547825179, 7.613462457402513, 8.042165076629360, 8.881475557679452, 10.62296992663263, 11.29976427066418, 12.70905976606632, 13.64913618416886, 14.79172056037881, 15.12474620287817, 16.20205000493621, 17.54613534341227, 18.51427961106496, 19.20853250133328

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