Properties

Label 2-101430-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s + 10-s − 6·13-s + 16-s + 2·17-s − 20-s + 23-s + 25-s + 6·26-s − 6·29-s − 8·31-s − 32-s − 2·34-s + 10·37-s + 40-s − 6·41-s − 8·43-s − 46-s + 8·47-s − 50-s − 6·52-s + 6·53-s + 6·58-s − 4·59-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 0.208·23-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 + 1.64·37-s + 0.158·40-s − 0.937·41-s − 1.21·43-s − 0.147·46-s + 1.16·47-s − 0.141·50-s − 0.832·52-s + 0.824·53-s + 0.787·58-s − 0.520·59-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: $\chi_{101430} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 101430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4861279516\)
\(L(\frac12)\) \(\approx\) \(0.4861279516\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.77457959509221, −13.07115688203561, −12.68264934199069, −12.24438001191652, −11.66343825777461, −11.32181899921255, −10.82305267431184, −10.08749292371411, −9.853969747140141, −9.335552241526568, −8.804788471260228, −8.233584776054083, −7.692009184517690, −7.233066249361155, −7.017516665652421, −6.214416289312326, −5.437549487209360, −5.236225568023602, −4.371120881507728, −3.839126961028055, −3.137673384077453, −2.524713803425435, −1.959376108488721, −1.169676411140891, −0.2577818088736915, 0.2577818088736915, 1.169676411140891, 1.959376108488721, 2.524713803425435, 3.137673384077453, 3.839126961028055, 4.371120881507728, 5.236225568023602, 5.437549487209360, 6.214416289312326, 7.017516665652421, 7.233066249361155, 7.692009184517690, 8.233584776054083, 8.804788471260228, 9.335552241526568, 9.853969747140141, 10.08749292371411, 10.82305267431184, 11.32181899921255, 11.66343825777461, 12.24438001191652, 12.68264934199069, 13.07115688203561, 13.77457959509221

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