Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13^{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-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 15-s + 16-s − 6·17-s − 18-s − 4·19-s + 20-s + 24-s + 25-s − 27-s + 6·29-s + 30-s − 4·31-s − 32-s + 6·34-s + 36-s + 10·37-s + 4·38-s − 40-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.64·37-s + 0.648·38-s − 0.158·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(248430\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{248430} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 248430,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−12.90369411146806, −12.58265429454467, −12.22936742635234, −11.51853373029175, −11.00911729467170, −10.85976025677367, −10.45223648297615, −9.780150025068219, −9.278398374630640, −9.099007877108904, −8.468092662126774, −7.907685225548964, −7.496455187701188, −6.882972010697271, −6.318039305125089, −6.187393754202973, −5.630403347913719, −4.800276461103827, −4.468927402422523, −3.997042172778481, −3.040772059509150, −2.546217856023699, −2.026729639018663, −1.396185333006164, −0.6872242756524382, 0, 0.6872242756524382, 1.396185333006164, 2.026729639018663, 2.546217856023699, 3.040772059509150, 3.997042172778481, 4.468927402422523, 4.800276461103827, 5.630403347913719, 6.187393754202973, 6.318039305125089, 6.882972010697271, 7.496455187701188, 7.907685225548964, 8.468092662126774, 9.099007877108904, 9.278398374630640, 9.780150025068219, 10.45223648297615, 10.85976025677367, 11.00911729467170, 11.51853373029175, 12.22936742635234, 12.58265429454467, 12.90369411146806

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