Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \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-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 12-s + 14-s − 15-s + 16-s + 6·17-s − 18-s + 4·19-s − 20-s − 21-s − 24-s + 25-s + 27-s − 28-s + 6·29-s + 30-s + 4·31-s − 32-s − 6·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.267·14-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 − 0.218·21-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.182·30-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(35490\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{35490} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 35490,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.394747010$
$L(\frac12)$  $\approx$  $2.394747010$
$L(\frac{3}{2})$   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 \( 1 + T \)
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

−14.88817970663635, −14.48810562655376, −14.01935365005574, −13.35972954832622, −12.77958953818142, −12.20128716691535, −11.78273344116878, −11.26294107048308, −10.48805628245388, −10.00252032875565, −9.623068665313788, −9.113820505010988, −8.266453374284025, −8.076830524929827, −7.495938912071249, −6.897466233117092, −6.332964258490978, −5.581494169145739, −4.954510775112699, −4.078662943282078, −3.470854198887857, −2.883751549661091, −2.312692018367656, −1.148434886143559, −0.7456128129022267, 0.7456128129022267, 1.148434886143559, 2.312692018367656, 2.883751549661091, 3.470854198887857, 4.078662943282078, 4.954510775112699, 5.581494169145739, 6.332964258490978, 6.897466233117092, 7.495938912071249, 8.076830524929827, 8.266453374284025, 9.113820505010988, 9.623068665313788, 10.00252032875565, 10.48805628245388, 11.26294107048308, 11.78273344116878, 12.20128716691535, 12.77958953818142, 13.35972954832622, 14.01935365005574, 14.48810562655376, 14.88817970663635

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