Properties

Label 2-286650-1.1-c1-0-137
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 − 6·11-s − 13-s + 16-s + 19-s + 6·22-s − 6·23-s + 26-s + 3·29-s − 8·31-s − 32-s − 37-s − 38-s − 9·41-s + 8·43-s − 6·44-s + 6·46-s − 3·47-s − 52-s − 3·53-s − 3·58-s − 6·59-s + 10·61-s + 8·62-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s − 0.277·13-s + 1/4·16-s + 0.229·19-s + 1.27·22-s − 1.25·23-s + 0.196·26-s + 0.557·29-s − 1.43·31-s − 0.176·32-s − 0.164·37-s − 0.162·38-s − 1.40·41-s + 1.21·43-s − 0.904·44-s + 0.884·46-s − 0.437·47-s − 0.138·52-s − 0.412·53-s − 0.393·58-s − 0.781·59-s + 1.28·61-s + 1.01·62-s + 1/8·64-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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.03442646786995, −12.31747036008475, −12.10284024079808, −11.56221968786957, −10.86833732873867, −10.64146759084043, −10.25805206925063, −9.747409773338561, −9.342340008634300, −8.763775852700085, −8.235420076373582, −7.877444595780707, −7.503558090910411, −7.043522778821281, −6.410262060404334, −5.864196279595900, −5.407854604258643, −4.971185832685207, −4.374276897972867, −3.626262226968822, −3.128149646781361, −2.531604345835929, −2.055477600828272, −1.509910161983639, −0.5320631161282894, 0, 0.5320631161282894, 1.509910161983639, 2.055477600828272, 2.531604345835929, 3.128149646781361, 3.626262226968822, 4.374276897972867, 4.971185832685207, 5.407854604258643, 5.864196279595900, 6.410262060404334, 7.043522778821281, 7.503558090910411, 7.877444595780707, 8.235420076373582, 8.763775852700085, 9.342340008634300, 9.747409773338561, 10.25805206925063, 10.64146759084043, 10.86833732873867, 11.56221968786957, 12.10284024079808, 12.31747036008475, 13.03442646786995

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