Properties

Label 2-259182-1.1-c1-0-138
Degree $2$
Conductor $259182$
Sign $-1$
Analytic cond. $2069.57$
Root an. cond. $45.4926$
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 + 2·5-s − 7-s + 8-s + 2·10-s + 2·13-s − 14-s + 16-s + 17-s − 4·19-s + 2·20-s − 8·23-s − 25-s + 2·26-s − 28-s + 6·29-s + 32-s + 34-s − 2·35-s − 2·37-s − 4·38-s + 2·40-s + 10·41-s + 4·43-s − 8·46-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.447·20-s − 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.11·29-s + 0.176·32-s + 0.171·34-s − 0.338·35-s − 0.328·37-s − 0.648·38-s + 0.316·40-s + 1.56·41-s + 0.609·43-s − 1.17·46-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(259182\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(2069.57\)
Root analytic conductor: \(45.4926\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 259182,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.07006918388857, −12.67326760471423, −12.14244841099974, −11.91973582311733, −11.16266295530157, −10.70333081171653, −10.38531764948565, −9.851397632365338, −9.433684130479214, −8.959854041877804, −8.209426259939156, −8.012556105552876, −7.301018678477284, −6.702932232553740, −6.205310496507083, −5.978056242790270, −5.578119503748232, −4.870114390189891, −4.227001711687739, −4.000081859308686, −3.247932946753808, −2.682010418579596, −2.160153326981869, −1.665816474681244, −0.9386104699532511, 0, 0.9386104699532511, 1.665816474681244, 2.160153326981869, 2.682010418579596, 3.247932946753808, 4.000081859308686, 4.227001711687739, 4.870114390189891, 5.578119503748232, 5.978056242790270, 6.205310496507083, 6.702932232553740, 7.301018678477284, 8.012556105552876, 8.209426259939156, 8.959854041877804, 9.433684130479214, 9.851397632365338, 10.38531764948565, 10.70333081171653, 11.16266295530157, 11.91973582311733, 12.14244841099974, 12.67326760471423, 13.07006918388857

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