Properties

Degree $2$
Conductor $49725$
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 − 4-s + 3·8-s − 4·11-s − 13-s − 16-s + 17-s − 4·19-s + 4·22-s + 26-s + 2·29-s − 8·31-s − 5·32-s − 34-s + 2·37-s + 4·38-s − 2·41-s + 4·43-s + 4·44-s + 8·47-s − 7·49-s + 52-s − 10·53-s − 2·58-s − 4·59-s + 14·61-s + 8·62-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.20·11-s − 0.277·13-s − 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.852·22-s + 0.196·26-s + 0.371·29-s − 1.43·31-s − 0.883·32-s − 0.171·34-s + 0.328·37-s + 0.648·38-s − 0.312·41-s + 0.609·43-s + 0.603·44-s + 1.16·47-s − 49-s + 0.138·52-s − 1.37·53-s − 0.262·58-s − 0.520·59-s + 1.79·61-s + 1.01·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49725\)    =    \(3^{2} \cdot 5^{2} \cdot 13 \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{49725} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 49725,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 + T \)
17 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.67562060585128, −14.32591770450159, −13.76561484650093, −13.14554230432895, −12.72203867347717, −12.49425489968195, −11.53352664025030, −10.95117649920616, −10.60067181122836, −10.09638945562727, −9.541732397761804, −9.101696934436715, −8.461120036400101, −7.996676996333804, −7.622364957390274, −6.979727694505295, −6.324742914295962, −5.442498989655577, −5.190142937470507, −4.437803937522181, −3.907947509775914, −3.107525282147906, −2.321915387944161, −1.716265495985696, −0.7120229379600252, 0, 0.7120229379600252, 1.716265495985696, 2.321915387944161, 3.107525282147906, 3.907947509775914, 4.437803937522181, 5.190142937470507, 5.442498989655577, 6.324742914295962, 6.979727694505295, 7.622364957390274, 7.996676996333804, 8.461120036400101, 9.101696934436715, 9.541732397761804, 10.09638945562727, 10.60067181122836, 10.95117649920616, 11.53352664025030, 12.49425489968195, 12.72203867347717, 13.14554230432895, 13.76561484650093, 14.32591770450159, 14.67562060585128

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