Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7^{2} \cdot 17 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5-s − 2·11-s + 5·13-s + 4·16-s + 17-s − 2·19-s + 2·20-s + 23-s + 25-s − 8·29-s − 31-s − 3·37-s − 7·41-s + 4·44-s − 47-s − 10·52-s + 8·53-s + 2·55-s + 7·61-s − 8·64-s − 5·65-s + 16·67-s − 2·68-s + 10·71-s + 8·73-s + 4·76-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s − 0.603·11-s + 1.38·13-s + 16-s + 0.242·17-s − 0.458·19-s + 0.447·20-s + 0.208·23-s + 1/5·25-s − 1.48·29-s − 0.179·31-s − 0.493·37-s − 1.09·41-s + 0.603·44-s − 0.145·47-s − 1.38·52-s + 1.09·53-s + 0.269·55-s + 0.896·61-s − 64-s − 0.620·65-s + 1.95·67-s − 0.242·68-s + 1.18·71-s + 0.936·73-s + 0.458·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37485 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37485 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(37485\)    =    \(3^{2} \cdot 5 \cdot 7^{2} \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{37485} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 37485,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.099376046\)
\(L(\frac12)\)  \(\approx\)  \(1.099376046\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
17 \( 1 - T \)
good2 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.76559252307448, −14.38937126322712, −13.66260594877840, −13.35480880389955, −12.81644143634773, −12.47512835407167, −11.63551261392338, −11.16702649684653, −10.66797344896095, −10.06364510947698, −9.543914591508268, −8.856518134481282, −8.430077765276156, −8.090243033214240, −7.359623617687321, −6.741390550824430, −5.994284498566726, −5.336107680500828, −5.046016483794456, −3.961086367130372, −3.848004082564933, −3.155649505761919, −2.149671258779065, −1.269458573596549, −0.4272856811233123, 0.4272856811233123, 1.269458573596549, 2.149671258779065, 3.155649505761919, 3.848004082564933, 3.961086367130372, 5.046016483794456, 5.336107680500828, 5.994284498566726, 6.741390550824430, 7.359623617687321, 8.090243033214240, 8.430077765276156, 8.856518134481282, 9.543914591508268, 10.06364510947698, 10.66797344896095, 11.16702649684653, 11.63551261392338, 12.47512835407167, 12.81644143634773, 13.35480880389955, 13.66260594877840, 14.38937126322712, 14.76559252307448

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