Properties

Degree $2$
Conductor $101430$
Sign $-1$
Motivic weight $1$
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 − 4·11-s + 2·13-s + 16-s − 6·17-s − 4·19-s + 20-s + 4·22-s + 23-s + 25-s − 2·26-s + 2·29-s − 32-s + 6·34-s − 2·37-s + 4·38-s − 40-s + 10·41-s − 4·43-s − 4·44-s − 46-s − 50-s + 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s − 0.392·26-s + 0.371·29-s − 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.158·40-s + 1.56·41-s − 0.609·43-s − 0.603·44-s − 0.147·46-s − 0.141·50-s + 0.277·52-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{101430} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−13.93057064782458, −13.37859355619611, −12.91468267376231, −12.77905836275113, −11.89866726739295, −11.39636001629596, −10.84456340661135, −10.56534127875724, −10.17915931462649, −9.416332450701335, −9.063210722319542, −8.507569211451803, −8.154358451278309, −7.534153066916943, −6.943134390118439, −6.463506738626397, −5.954989053833726, −5.448402166120509, −4.635819178428376, −4.329695287419866, −3.355414779245549, −2.766039438793998, −2.187874740955043, −1.716953921282689, −0.7431787235417628, 0, 0.7431787235417628, 1.716953921282689, 2.187874740955043, 2.766039438793998, 3.355414779245549, 4.329695287419866, 4.635819178428376, 5.448402166120509, 5.954989053833726, 6.463506738626397, 6.943134390118439, 7.534153066916943, 8.154358451278309, 8.507569211451803, 9.063210722319542, 9.416332450701335, 10.17915931462649, 10.56534127875724, 10.84456340661135, 11.39636001629596, 11.89866726739295, 12.77905836275113, 12.91468267376231, 13.37859355619611, 13.93057064782458

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