Properties

Degree 2
Conductor $ 2^{3} \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  + 4·7-s − 3·9-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s − 4·23-s − 2·29-s − 8·31-s − 6·37-s − 6·41-s + 8·43-s − 4·47-s + 9·49-s − 6·53-s − 4·59-s − 2·61-s − 12·63-s − 8·67-s + 6·73-s + 16·77-s + 9·81-s + 16·83-s − 6·89-s + 8·91-s + 14·97-s − 12·99-s + ⋯
L(s)  = 1  + 1.51·7-s − 9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s − 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s − 0.256·61-s − 1.51·63-s − 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s + 1.75·83-s − 0.635·89-s + 0.838·91-s + 1.42·97-s − 1.20·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(200\)    =    \(2^{3} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 200,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.327698878$
$L(\frac12)$  $\approx$  $1.327698878$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−19.71415970002014, −18.46229925374627, −17.71991267261816, −17.13303675675406, −16.14657841475887, −14.95976150980043, −14.28069790269282, −13.71772139542571, −12.16106454520843, −11.45666669128344, −10.83635878381862, −9.267253735733507, −8.516348691113562, −7.509321760734207, −6.111657480149244, −5.019785669158583, −3.667959270251673, −1.731858114678643, 1.731858114678643, 3.667959270251673, 5.019785669158583, 6.111657480149244, 7.509321760734207, 8.516348691113562, 9.267253735733507, 10.83635878381862, 11.45666669128344, 12.16106454520843, 13.71772139542571, 14.28069790269282, 14.95976150980043, 16.14657841475887, 17.13303675675406, 17.71991267261816, 18.46229925374627, 19.71415970002014

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