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Properties

Label 2-59290-1.1-c1-0-119
Degree $2$
Conductor $59290$
Sign $1$
Analytic cond. $473.433$
Root an. cond. $21.7585$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 3·9-s − 10-s − 6·13-s + 16-s + 2·17-s + 3·18-s + 20-s + 25-s + 6·26-s − 6·29-s − 8·31-s − 32-s − 2·34-s − 3·36-s − 10·37-s − 40-s + 2·41-s − 4·43-s − 3·45-s − 8·47-s − 50-s − 6·52-s − 2·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 9-s − 0.316·10-s − 1.66·13-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.223·20-s + 1/5·25-s + 1.17·26-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 1/2·36-s − 1.64·37-s − 0.158·40-s + 0.312·41-s − 0.609·43-s − 0.447·45-s − 1.16·47-s − 0.141·50-s − 0.832·52-s − 0.274·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(59290\)    =    \(2 \cdot 5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(473.433\)
Root analytic conductor: \(21.7585\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 59290,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
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

−14.83234628310706, −14.57594595951272, −13.81915740695851, −13.40841952219810, −12.62960892727948, −12.22643163176630, −11.80404115729863, −11.20074490674007, −10.64996661449657, −10.24769858487310, −9.500826864139895, −9.344538776060250, −8.722039888134858, −8.134171471902363, −7.493760846186632, −7.202243649224968, −6.493852757742437, −5.840624894260170, −5.293994311733450, −4.972525393026807, −3.945221969133152, −3.186366346095267, −2.728119826532394, −1.956124455637151, −1.469479087900107, 0, 0, 1.469479087900107, 1.956124455637151, 2.728119826532394, 3.186366346095267, 3.945221969133152, 4.972525393026807, 5.293994311733450, 5.840624894260170, 6.493852757742437, 7.202243649224968, 7.493760846186632, 8.134171471902363, 8.722039888134858, 9.344538776060250, 9.500826864139895, 10.24769858487310, 10.64996661449657, 11.20074490674007, 11.80404115729863, 12.22643163176630, 12.62960892727948, 13.40841952219810, 13.81915740695851, 14.57594595951272, 14.83234628310706

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