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 1

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 − 8·19-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 − 1.83·19-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  =  \(1\)
Selberg data  =  \((2,\ 35490,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + T \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.09625664170999, −14.70892160698242, −14.20483979292938, −13.48102055312403, −13.19119118187986, −12.64336885875489, −12.29506556865520, −11.35109126822143, −11.22403610545155, −10.39451926479543, −10.04825183849404, −9.155666619600774, −8.733773106673326, −8.177569407494123, −7.631139461152238, −6.889618384297478, −6.312393638884782, −6.152747051450104, −4.902625909756621, −4.497459875418853, −4.058375999450031, −3.351609796494592, −2.451968816860291, −2.324338764204478, −1.110288851565094, 0, 1.110288851565094, 2.324338764204478, 2.451968816860291, 3.351609796494592, 4.058375999450031, 4.497459875418853, 4.902625909756621, 6.152747051450104, 6.312393638884782, 6.889618384297478, 7.631139461152238, 8.177569407494123, 8.733773106673326, 9.155666619600774, 10.04825183849404, 10.39451926479543, 11.22403610545155, 11.35109126822143, 12.29506556865520, 12.64336885875489, 13.19119118187986, 13.48102055312403, 14.20483979292938, 14.70892160698242, 15.09625664170999

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