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Properties

Label 2-101430-1.1-c1-0-121
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 + 5·13-s + 16-s − 19-s − 20-s − 23-s + 25-s + 5·26-s + 3·29-s − 7·31-s + 32-s + 8·37-s − 38-s − 40-s + 6·41-s − 4·43-s − 46-s − 12·47-s + 50-s + 5·52-s − 9·53-s + 3·58-s + 3·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 1.38·13-s + 1/4·16-s − 0.229·19-s − 0.223·20-s − 0.208·23-s + 1/5·25-s + 0.980·26-s + 0.557·29-s − 1.25·31-s + 0.176·32-s + 1.31·37-s − 0.162·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s − 0.147·46-s − 1.75·47-s + 0.141·50-s + 0.693·52-s − 1.23·53-s + 0.393·58-s + 0.390·59-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 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−14.06326946962607, −13.31254926920448, −13.06499236039234, −12.70771449811501, −11.98381177716960, −11.56345021559440, −11.14600810207226, −10.72501487336152, −10.22924064157690, −9.480487519480652, −9.039975099971329, −8.412110104688668, −7.904160208545050, −7.548441682986926, −6.757885239395030, −6.286995672016373, −5.968852261527676, −5.230884937318506, −4.656071220733504, −4.160949933294797, −3.538059357237727, −3.189472067327333, −2.396113940440291, −1.641210015762745, −1.031155790392657, 0, 1.031155790392657, 1.641210015762745, 2.396113940440291, 3.189472067327333, 3.538059357237727, 4.160949933294797, 4.656071220733504, 5.230884937318506, 5.968852261527676, 6.286995672016373, 6.757885239395030, 7.548441682986926, 7.904160208545050, 8.412110104688668, 9.039975099971329, 9.480487519480652, 10.22924064157690, 10.72501487336152, 11.14600810207226, 11.56345021559440, 11.98381177716960, 12.70771449811501, 13.06499236039234, 13.31254926920448, 14.06326946962607

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