Properties

Degree $2$
Conductor $25350$
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 + 6-s + 8-s + 9-s − 4·11-s + 12-s + 16-s + 6·17-s + 18-s − 4·19-s − 4·22-s − 8·23-s + 24-s + 27-s + 6·29-s + 8·31-s + 32-s − 4·33-s + 6·34-s + 36-s − 10·37-s − 4·38-s + 6·41-s − 4·43-s − 4·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + 1.02·34-s + 1/6·36-s − 1.64·37-s − 0.648·38-s + 0.937·41-s − 0.609·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{25350} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25350,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.60266757844918, −15.04199786416887, −14.40975621973169, −14.03807446263427, −13.56553255098709, −13.02058904086203, −12.37231074155717, −12.08626835931234, −11.47349245587686, −10.50625613902698, −10.16469914353556, −9.982079312335805, −8.842375612299158, −8.283725617222704, −7.924675752999207, −7.298171257049364, −6.616914557782054, −5.845637967284277, −5.511647833112978, −4.518833868031745, −4.278596504232596, −3.214528649235723, −2.905543577418157, −2.102842924927731, −1.321791979833916, 0, 1.321791979833916, 2.102842924927731, 2.905543577418157, 3.214528649235723, 4.278596504232596, 4.518833868031745, 5.511647833112978, 5.845637967284277, 6.616914557782054, 7.298171257049364, 7.924675752999207, 8.283725617222704, 8.842375612299158, 9.982079312335805, 10.16469914353556, 10.50625613902698, 11.47349245587686, 12.08626835931234, 12.37231074155717, 13.02058904086203, 13.56553255098709, 14.03807446263427, 14.40975621973169, 15.04199786416887, 15.60266757844918

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