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Properties

Label 2-101430-1.1-c1-0-109
Degree $2$
Conductor $101430$
Sign $-1$
Analytic cond. $809.922$
Root an. cond. $28.4591$
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 + 5-s − 8-s − 10-s + 2·11-s − 2·13-s + 16-s − 2·17-s + 4·19-s + 20-s − 2·22-s − 23-s + 25-s + 2·26-s + 8·29-s − 2·31-s − 32-s + 2·34-s + 4·37-s − 4·38-s − 40-s − 2·41-s − 8·43-s + 2·44-s + 46-s − 12·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 0.603·11-s − 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.223·20-s − 0.426·22-s − 0.208·23-s + 1/5·25-s + 0.392·26-s + 1.48·29-s − 0.359·31-s − 0.176·32-s + 0.342·34-s + 0.657·37-s − 0.648·38-s − 0.158·40-s − 0.312·41-s − 1.21·43-s + 0.301·44-s + 0.147·46-s − 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(101430\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(809.922\)
Root analytic conductor: \(28.4591\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 101430,\ (\ :1/2),\ -1)\)

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

−14.07493602030145, −13.40378293716829, −13.13263816538322, −12.35295105667702, −11.92294241625173, −11.56549582714009, −11.04484385722368, −10.35203226047727, −9.998559469862018, −9.557569930555553, −9.154813002284385, −8.407092990271698, −8.229526712448198, −7.461700040625049, −6.865131715617795, −6.601216325190741, −6.000341457715749, −5.247038886528772, −4.906858830664691, −4.102240354632410, −3.454360657916372, −2.774546800698699, −2.247998127835241, −1.499821711594536, −0.9330568424803719, 0, 0.9330568424803719, 1.499821711594536, 2.247998127835241, 2.774546800698699, 3.454360657916372, 4.102240354632410, 4.906858830664691, 5.247038886528772, 6.000341457715749, 6.601216325190741, 6.865131715617795, 7.461700040625049, 8.229526712448198, 8.407092990271698, 9.154813002284385, 9.557569930555553, 9.998559469862018, 10.35203226047727, 11.04484385722368, 11.56549582714009, 11.92294241625173, 12.35295105667702, 13.13263816538322, 13.40378293716829, 14.07493602030145

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