Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s + 2·13-s − 6·17-s + 4·19-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 + 12·51-s + 6·53-s − 8·57-s − 12·59-s − 2·61-s − 2·67-s + 12·69-s − 12·71-s + 2·73-s + 8·79-s − 11·81-s + 6·83-s − 12·87-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 0.554·13-s − 1.45·17-s + 0.917·19-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 + 1.68·51-s + 0.824·53-s − 1.05·57-s − 1.56·59-s − 0.256·61-s − 0.244·67-s + 1.44·69-s − 1.42·71-s + 0.234·73-s + 0.900·79-s − 1.22·81-s + 0.658·83-s − 1.28·87-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4900,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
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

−17.84529825217496, −17.69635948760545, −16.87832112678091, −16.26241358430512, −15.75910015274063, −15.35226321862306, −14.26035098913108, −13.78998221834218, −13.19750290854228, −12.29735580113336, −11.86256892092283, −11.36536247196132, −10.61881182362287, −10.23587508712658, −9.278509798118928, −8.619355787860042, −7.884192193406701, −6.975510920006742, −6.278201721491214, −5.907041679865618, −4.924044727566885, −4.428548672318472, −3.377833180183632, −2.329798435607222, −1.137819692591272, 0, 1.137819692591272, 2.329798435607222, 3.377833180183632, 4.428548672318472, 4.924044727566885, 5.907041679865618, 6.278201721491214, 6.975510920006742, 7.884192193406701, 8.619355787860042, 9.278509798118928, 10.23587508712658, 10.61881182362287, 11.36536247196132, 11.86256892092283, 12.29735580113336, 13.19750290854228, 13.78998221834218, 14.26035098913108, 15.35226321862306, 15.75910015274063, 16.26241358430512, 16.87832112678091, 17.69635948760545, 17.84529825217496

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