Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} \cdot 7^{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  − 3-s − 2·9-s − 2·11-s + 4·17-s − 2·19-s + 23-s + 5·27-s + 9·29-s + 4·31-s + 2·33-s − 4·37-s − 41-s + 9·43-s − 4·51-s + 10·53-s + 2·57-s − 10·59-s − 9·61-s + 5·67-s − 69-s − 14·71-s + 12·73-s − 14·79-s + 81-s − 11·83-s − 9·87-s + 15·89-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 0.603·11-s + 0.970·17-s − 0.458·19-s + 0.208·23-s + 0.962·27-s + 1.67·29-s + 0.718·31-s + 0.348·33-s − 0.657·37-s − 0.156·41-s + 1.37·43-s − 0.560·51-s + 1.37·53-s + 0.264·57-s − 1.30·59-s − 1.15·61-s + 0.610·67-s − 0.120·69-s − 1.66·71-s + 1.40·73-s − 1.57·79-s + 1/9·81-s − 1.20·83-s − 0.964·87-s + 1.58·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.320417990$
$L(\frac12)$  $\approx$  $1.320417990$
$L(\frac{3}{2})$   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 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.69052764235171, −15.27723551557995, −14.50578643904909, −14.04785906467713, −13.62765393731980, −12.80976683295856, −12.26856222684206, −11.94827820254114, −11.28385592834921, −10.58585764145155, −10.35404655289351, −9.660188420438223, −8.770263125378468, −8.467397520737006, −7.719414694752442, −7.157435292349812, −6.286971922781291, −5.967290579369462, −5.203428928217074, −4.745980370783833, −3.923226928673195, −2.943942792321344, −2.608585801407894, −1.377558841890280, −0.5190751144666104, 0.5190751144666104, 1.377558841890280, 2.608585801407894, 2.943942792321344, 3.923226928673195, 4.745980370783833, 5.203428928217074, 5.967290579369462, 6.286971922781291, 7.157435292349812, 7.719414694752442, 8.467397520737006, 8.770263125378468, 9.660188420438223, 10.35404655289351, 10.58585764145155, 11.28385592834921, 11.94827820254114, 12.26856222684206, 12.80976683295856, 13.62765393731980, 14.04785906467713, 14.50578643904909, 15.27723551557995, 15.69052764235171

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