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Properties

Label 2-169050-1.1-c1-0-204
Degree $2$
Conductor $169050$
Sign $-1$
Analytic cond. $1349.87$
Root an. cond. $36.7405$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 2·11-s + 12-s + 4·13-s + 16-s − 6·17-s + 18-s + 4·19-s − 2·22-s + 23-s + 24-s + 4·26-s + 27-s + 2·29-s − 2·31-s + 32-s − 2·33-s − 6·34-s + 36-s − 4·37-s + 4·38-s + 4·39-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 − 0.603·11-s + 0.288·12-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.426·22-s + 0.208·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.371·29-s − 0.359·31-s + 0.176·32-s − 0.348·33-s − 1.02·34-s + 1/6·36-s − 0.657·37-s + 0.648·38-s + 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(1349.87\)
Root analytic conductor: \(36.7405\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 169050,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
23 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.42749014369042, −13.05858707887174, −12.79949624840380, −12.07646400570517, −11.54015160577063, −11.16048989501295, −10.76857222930404, −10.07387121919366, −9.820517363935662, −8.976944466948093, −8.586206506898393, −8.347697069133217, −7.486680027633747, −7.192340422479368, −6.623698767549442, −6.143577927325673, −5.460085702707110, −5.120539944315511, −4.385795748962391, −3.984585804879189, −3.402494178475210, −2.855019913099265, −2.352734051776937, −1.665461391185060, −1.048426167837874, 0, 1.048426167837874, 1.665461391185060, 2.352734051776937, 2.855019913099265, 3.402494178475210, 3.984585804879189, 4.385795748962391, 5.120539944315511, 5.460085702707110, 6.143577927325673, 6.623698767549442, 7.192340422479368, 7.486680027633747, 8.347697069133217, 8.586206506898393, 8.976944466948093, 9.820517363935662, 10.07387121919366, 10.76857222930404, 11.16048989501295, 11.54015160577063, 12.07646400570517, 12.79949624840380, 13.05858707887174, 13.42749014369042

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