Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 13 \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  − 2·7-s + 13-s + 17-s + 8·19-s + 6·23-s + 7·29-s + 3·31-s + 2·37-s + 12·41-s + 8·43-s − 6·47-s − 3·49-s − 9·53-s − 9·59-s + 2·61-s − 67-s − 2·71-s − 2·73-s − 6·79-s + 12·83-s − 89-s − 2·91-s − 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.277·13-s + 0.242·17-s + 1.83·19-s + 1.25·23-s + 1.29·29-s + 0.538·31-s + 0.328·37-s + 1.87·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s − 1.23·53-s − 1.17·59-s + 0.256·61-s − 0.122·67-s − 0.237·71-s − 0.234·73-s − 0.675·79-s + 1.31·83-s − 0.105·89-s − 0.209·91-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(397800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{397800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 397800,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.062011126$
$L(\frac12)$  $\approx$  $3.062011126$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−12.46586855616870, −12.00298423861138, −11.58026602835247, −11.02567498102914, −10.72444372505263, −10.14492804161084, −9.591763460558755, −9.342958951751219, −9.046741922067933, −8.214734264492319, −7.919150394535738, −7.387813819339338, −6.964092116254019, −6.363525026574557, −6.078944855941612, −5.480084382174023, −4.949864002836018, −4.535621762025264, −3.887912659308609, −3.277557156411874, −2.845710843560881, −2.623171951634170, −1.474247924398358, −1.092744880544564, −0.5076240489910886, 0.5076240489910886, 1.092744880544564, 1.474247924398358, 2.623171951634170, 2.845710843560881, 3.277557156411874, 3.887912659308609, 4.535621762025264, 4.949864002836018, 5.480084382174023, 6.078944855941612, 6.363525026574557, 6.964092116254019, 7.387813819339338, 7.919150394535738, 8.214734264492319, 9.046741922067933, 9.342958951751219, 9.591763460558755, 10.14492804161084, 10.72444372505263, 11.02567498102914, 11.58026602835247, 12.00298423861138, 12.46586855616870

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