Properties

Degree 2
Conductor $ 5^{2} \cdot 7 \cdot 23 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s + 7-s − 2·9-s − 11-s + 2·12-s − 13-s + 4·16-s − 17-s + 2·19-s − 21-s − 23-s + 5·27-s − 2·28-s + 7·29-s + 4·31-s + 33-s + 4·36-s − 8·37-s + 39-s − 6·41-s + 8·43-s + 2·44-s + 7·47-s − 4·48-s + 49-s + 51-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.377·7-s − 2/3·9-s − 0.301·11-s + 0.577·12-s − 0.277·13-s + 16-s − 0.242·17-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.962·27-s − 0.377·28-s + 1.29·29-s + 0.718·31-s + 0.174·33-s + 2/3·36-s − 1.31·37-s + 0.160·39-s − 0.937·41-s + 1.21·43-s + 0.301·44-s + 1.02·47-s − 0.577·48-s + 1/7·49-s + 0.140·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4025} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4025,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−8.227424872801678942008594600403, −7.44488666107612493453369797646, −6.46305776873434412629623573176, −5.70551226127103814314548639498, −5.02837085429780677206654130848, −4.53638690439984486481221985630, −3.48098161163945720129544839182, −2.55208017739021272500029411580, −1.11047280102303688568605393583, 0, 1.11047280102303688568605393583, 2.55208017739021272500029411580, 3.48098161163945720129544839182, 4.53638690439984486481221985630, 5.02837085429780677206654130848, 5.70551226127103814314548639498, 6.46305776873434412629623573176, 7.44488666107612493453369797646, 8.227424872801678942008594600403

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