Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 167 $
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 − 4·7-s + 9-s + 4·11-s + 4·17-s + 4·19-s − 4·21-s − 4·23-s + 27-s − 2·29-s − 4·31-s + 4·33-s + 12·37-s + 12·41-s − 8·43-s + 9·49-s + 4·51-s − 14·53-s + 4·57-s − 2·59-s − 2·61-s − 4·63-s − 4·67-s − 4·69-s − 8·71-s − 6·73-s − 16·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.970·17-s + 0.917·19-s − 0.872·21-s − 0.834·23-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.696·33-s + 1.97·37-s + 1.87·41-s − 1.21·43-s + 9/7·49-s + 0.560·51-s − 1.92·53-s + 0.529·57-s − 0.260·59-s − 0.256·61-s − 0.503·63-s − 0.488·67-s − 0.481·69-s − 0.949·71-s − 0.702·73-s − 1.82·77-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 200400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 200400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(200400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{200400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 200400,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.578307447$
$L(\frac12)$  $\approx$  $2.578307447$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
167 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−13.21578631643929, −12.55446353464498, −12.19739127672469, −11.83485661465523, −11.16216701931302, −10.71166785131426, −9.982526252163350, −9.611772865755629, −9.323062214755505, −9.154373647522266, −8.181051337613743, −7.824014714760739, −7.397241414817767, −6.712757487089218, −6.380665935339169, −5.858281834871745, −5.438802604195094, −4.506484038097923, −4.015894944362860, −3.616179977341044, −2.980285457281289, −2.759291116401861, −1.739906106589661, −1.238704396505696, −0.4526849861750042, 0.4526849861750042, 1.238704396505696, 1.739906106589661, 2.759291116401861, 2.980285457281289, 3.616179977341044, 4.015894944362860, 4.506484038097923, 5.438802604195094, 5.858281834871745, 6.380665935339169, 6.712757487089218, 7.397241414817767, 7.824014714760739, 8.181051337613743, 9.154373647522266, 9.323062214755505, 9.611772865755629, 9.982526252163350, 10.71166785131426, 11.16216701931302, 11.83485661465523, 12.19739127672469, 12.55446353464498, 13.21578631643929

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