Properties

Label 2-101430-1.1-c1-0-80
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 + 2·11-s + 6·13-s + 16-s + 6·17-s − 20-s + 2·22-s + 23-s + 25-s + 6·26-s − 4·29-s + 2·31-s + 32-s + 6·34-s + 4·37-s − 40-s + 2·41-s − 4·43-s + 2·44-s + 46-s + 50-s + 6·52-s + 6·53-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 + 1.66·13-s + 1/4·16-s + 1.45·17-s − 0.223·20-s + 0.426·22-s + 0.208·23-s + 1/5·25-s + 1.17·26-s − 0.742·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s + 0.657·37-s − 0.158·40-s + 0.312·41-s − 0.609·43-s + 0.301·44-s + 0.147·46-s + 0.141·50-s + 0.832·52-s + 0.824·53-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

\(L(1)\) \(\approx\) \(5.774899577\)
\(L(\frac12)\) \(\approx\) \(5.774899577\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.71848994572153, −13.31206831164936, −12.72926610963797, −12.37474939594495, −11.72284011885073, −11.41227778596955, −10.99228902365857, −10.42220862808666, −9.831925175550393, −9.334608916114962, −8.614028131611417, −8.266416622119172, −7.668974774742072, −7.197512983875948, −6.428261066387539, −6.229232779524173, −5.466230134066216, −5.148533973807605, −4.263213654048801, −3.759611692979731, −3.522183083423491, −2.825991595819429, −1.986991038417839, −1.209508588644138, −0.7595043952304482, 0.7595043952304482, 1.209508588644138, 1.986991038417839, 2.825991595819429, 3.522183083423491, 3.759611692979731, 4.263213654048801, 5.148533973807605, 5.466230134066216, 6.229232779524173, 6.428261066387539, 7.197512983875948, 7.668974774742072, 8.266416622119172, 8.614028131611417, 9.334608916114962, 9.831925175550393, 10.42220862808666, 10.99228902365857, 11.41227778596955, 11.72284011885073, 12.37474939594495, 12.72926610963797, 13.31206831164936, 13.71848994572153

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