Properties

Label 2-286650-1.1-c1-0-257
Degree $2$
Conductor $286650$
Sign $-1$
Analytic cond. $2288.91$
Root an. cond. $47.8425$
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 + 4-s − 8-s − 5·11-s + 13-s + 16-s − 17-s + 19-s + 5·22-s + 5·23-s − 26-s − 29-s + 6·31-s − 32-s + 34-s − 7·37-s − 38-s + 2·41-s − 43-s − 5·44-s − 5·46-s + 52-s + 6·53-s + 58-s + 6·59-s + 7·61-s − 6·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.50·11-s + 0.277·13-s + 1/4·16-s − 0.242·17-s + 0.229·19-s + 1.06·22-s + 1.04·23-s − 0.196·26-s − 0.185·29-s + 1.07·31-s − 0.176·32-s + 0.171·34-s − 1.15·37-s − 0.162·38-s + 0.312·41-s − 0.152·43-s − 0.753·44-s − 0.737·46-s + 0.138·52-s + 0.824·53-s + 0.131·58-s + 0.781·59-s + 0.896·61-s − 0.762·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(286650\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(2288.91\)
Root analytic conductor: \(47.8425\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 286650,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.91409552176128, −12.57776290855119, −11.87717240672082, −11.55175064074487, −10.95930580423900, −10.67897936711104, −10.11312595498392, −9.907860279836660, −9.205966744787182, −8.749856061104868, −8.333093650527086, −7.961712981930255, −7.358051326260487, −6.998298432355191, −6.523291605294159, −5.851357430992482, −5.366738930913121, −4.988223106316718, −4.385056798563589, −3.589797678025667, −3.167068478388943, −2.449385299385054, −2.223894449681386, −1.287531737890300, −0.7332639964831605, 0, 0.7332639964831605, 1.287531737890300, 2.223894449681386, 2.449385299385054, 3.167068478388943, 3.589797678025667, 4.385056798563589, 4.988223106316718, 5.366738930913121, 5.851357430992482, 6.523291605294159, 6.998298432355191, 7.358051326260487, 7.961712981930255, 8.333093650527086, 8.749856061104868, 9.205966744787182, 9.907860279836660, 10.11312595498392, 10.67897936711104, 10.95930580423900, 11.55175064074487, 11.87717240672082, 12.57776290855119, 12.91409552176128

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