Properties

Label 2-381150-1.1-c1-0-433
Degree $2$
Conductor $381150$
Sign $1$
Analytic cond. $3043.49$
Root an. cond. $55.1679$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s − 4·13-s − 14-s + 16-s + 4·19-s + 4·26-s + 28-s − 6·29-s − 10·31-s − 32-s − 2·37-s − 4·38-s − 12·41-s − 4·43-s + 6·47-s + 49-s − 4·52-s − 6·53-s − 56-s + 6·58-s + 6·59-s + 4·61-s + 10·62-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.917·19-s + 0.784·26-s + 0.188·28-s − 1.11·29-s − 1.79·31-s − 0.176·32-s − 0.328·37-s − 0.648·38-s − 1.87·41-s − 0.609·43-s + 0.875·47-s + 1/7·49-s − 0.554·52-s − 0.824·53-s − 0.133·56-s + 0.787·58-s + 0.781·59-s + 0.512·61-s + 1.27·62-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(381150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3043.49\)
Root analytic conductor: \(55.1679\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 381150,\ (\ :1/2),\ 1)\)

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 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.89287597150328, −12.30088641467759, −11.97709611829003, −11.50491974402947, −11.11979551672191, −10.67983448942469, −10.06462701363990, −9.767640606524070, −9.407405545586981, −8.741713716001662, −8.538739096122949, −7.833517131211414, −7.455844266476187, −7.036421551904327, −6.802010619005555, −5.894186945857543, −5.348185134518645, −5.299955815796129, −4.469330086284424, −3.908486707276805, −3.304377858609090, −2.839471710060365, −2.125892003544475, −1.676047419082754, −1.172861417169623, 0, 0, 1.172861417169623, 1.676047419082754, 2.125892003544475, 2.839471710060365, 3.304377858609090, 3.908486707276805, 4.469330086284424, 5.299955815796129, 5.348185134518645, 5.894186945857543, 6.802010619005555, 7.036421551904327, 7.455844266476187, 7.833517131211414, 8.538739096122949, 8.741713716001662, 9.407405545586981, 9.767640606524070, 10.06462701363990, 10.67983448942469, 11.11979551672191, 11.50491974402947, 11.97709611829003, 12.30088641467759, 12.89287597150328

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