Properties

Degree 2
Conductor $ 2^{2} \cdot 5^{2} $
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 − 2·7-s + 9-s − 2·13-s + 6·17-s − 4·19-s − 4·21-s − 6·23-s − 4·27-s + 6·29-s − 4·31-s − 2·37-s − 4·39-s + 6·41-s + 10·43-s + 6·47-s − 3·49-s + 12·51-s + 6·53-s − 8·57-s + 12·59-s + 2·61-s − 2·63-s − 2·67-s − 12·69-s − 12·71-s − 2·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 1/3·9-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.872·21-s − 1.25·23-s − 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.640·39-s + 0.937·41-s + 1.52·43-s + 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s − 1.05·57-s + 1.56·59-s + 0.256·61-s − 0.251·63-s − 0.244·67-s − 1.44·69-s − 1.42·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100\)    =    \(2^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 100,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.263098962$
$L(\frac12)$  $\approx$  $1.263098962$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.37702400620986, −19.23339240606807, −17.84322890580079, −16.67808032918220, −15.77984643452670, −14.62109083140550, −14.10055486094281, −12.92148359796848, −12.03947269145467, −10.36534498208387, −9.493169657561115, −8.414042470938793, −7.384090002225106, −5.857403178467226, −3.900749965053077, −2.583212561785407, 2.583212561785407, 3.900749965053077, 5.857403178467226, 7.384090002225106, 8.414042470938793, 9.493169657561115, 10.36534498208387, 12.03947269145467, 12.92148359796848, 14.10055486094281, 14.62109083140550, 15.77984643452670, 16.67808032918220, 17.84322890580079, 19.23339240606807, 19.37702400620986

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