Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{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 − 2·13-s − 14-s + 15-s + 16-s + 6·17-s + 18-s − 8·19-s + 20-s − 21-s + 24-s + 25-s − 2·26-s + 27-s − 28-s − 6·29-s + 30-s − 4·31-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 − 0.554·13-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 − 0.392·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.182·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(25410\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{25410} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 25410,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, 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 \( 1 + T \)
11 \( 1 \)
good13 \( 1 + 2 T + p T^{2} \)
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

−15.51822889178444, −14.80754382297840, −14.59007570011635, −14.07175909619821, −13.53730685971906, −12.86838627161426, −12.42793366255372, −12.31032055354493, −11.16423729507884, −10.81347074324359, −10.14609638475495, −9.629047360509960, −9.117986192177286, −8.378649876091768, −7.814464638914080, −7.147851538056561, −6.681631983856221, −5.873500074353444, −5.491190103719801, −4.720499901851684, −3.969193663237434, −3.487467229846676, −2.712218836751281, −2.096453512901847, −1.391546104420324, 0, 1.391546104420324, 2.096453512901847, 2.712218836751281, 3.487467229846676, 3.969193663237434, 4.720499901851684, 5.491190103719801, 5.873500074353444, 6.681631983856221, 7.147851538056561, 7.814464638914080, 8.378649876091768, 9.117986192177286, 9.629047360509960, 10.14609638475495, 10.81347074324359, 11.16423729507884, 12.31032055354493, 12.42793366255372, 12.86838627161426, 13.53730685971906, 14.07175909619821, 14.59007570011635, 14.80754382297840, 15.51822889178444

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