Properties

Degree 2
Conductor $ 2^{2} \cdot 5^{2} \cdot 13^{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  + 2·3-s + 2·7-s + 9-s + 6·17-s + 4·19-s + 4·21-s − 6·23-s − 4·27-s + 6·29-s + 4·31-s + 2·37-s − 6·41-s + 10·43-s − 6·47-s − 3·49-s + 12·51-s + 6·53-s + 8·57-s − 12·59-s + 2·61-s + 2·63-s + 2·67-s − 12·69-s + 12·71-s + 2·73-s + 8·79-s − 11·81-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 1/3·9-s + 1.45·17-s + 0.917·19-s + 0.872·21-s − 1.25·23-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s + 1.05·57-s − 1.56·59-s + 0.256·61-s + 0.251·63-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 + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16900\)    =    \(2^{2} \cdot 5^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{16900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 16900,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.203847453$
$L(\frac12)$  $\approx$  $4.203847453$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
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

−15.78812784542964, −15.25115479403371, −14.62218971780950, −14.17465762398010, −13.92325064422156, −13.39824380257110, −12.47075703141230, −12.03043126594466, −11.56227037346100, −10.78017453829248, −10.07256869843619, −9.666466810151499, −9.058468071259215, −8.272247941931688, −7.935122978007670, −7.640589517983138, −6.677792075393693, −5.920253280333684, −5.254253030363432, −4.578009367471638, −3.736117493923519, −3.196051486894461, −2.487838720123482, −1.707430277054586, −0.8536938349441409, 0.8536938349441409, 1.707430277054586, 2.487838720123482, 3.196051486894461, 3.736117493923519, 4.578009367471638, 5.254253030363432, 5.920253280333684, 6.677792075393693, 7.640589517983138, 7.935122978007670, 8.272247941931688, 9.058468071259215, 9.666466810151499, 10.07256869843619, 10.78017453829248, 11.56227037346100, 12.03043126594466, 12.47075703141230, 13.39824380257110, 13.92325064422156, 14.17465762398010, 14.62218971780950, 15.25115479403371, 15.78812784542964

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