Properties

Label 2-259182-1.1-c1-0-122
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 − 5-s + 7-s + 8-s − 10-s + 2·13-s + 14-s + 16-s − 17-s + 5·19-s − 20-s − 5·23-s − 4·25-s + 2·26-s + 28-s − 2·29-s − 31-s + 32-s − 34-s − 35-s + 37-s + 5·38-s − 40-s + 6·41-s − 12·43-s − 5·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.14·19-s − 0.223·20-s − 1.04·23-s − 4/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.179·31-s + 0.176·32-s − 0.171·34-s − 0.169·35-s + 0.164·37-s + 0.811·38-s − 0.158·40-s + 0.937·41-s − 1.82·43-s − 0.737·46-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 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−12.92894208069401, −12.79936644921481, −11.88387178891786, −11.73761389920870, −11.44369050298382, −10.93174804156257, −10.26115505119941, −9.949816856806844, −9.398787253145600, −8.778111585409567, −8.187787464561668, −7.973566175466936, −7.253384129321391, −7.062308160495767, −6.226342157283219, −5.923855884577349, −5.344681113576548, −4.928364948075899, −4.223783677598240, −3.896105704207616, −3.392009912390269, −2.822262857757904, −2.072116130754741, −1.620355833011532, −0.8728409970989701, 0, 0.8728409970989701, 1.620355833011532, 2.072116130754741, 2.822262857757904, 3.392009912390269, 3.896105704207616, 4.223783677598240, 4.928364948075899, 5.344681113576548, 5.923855884577349, 6.226342157283219, 7.062308160495767, 7.253384129321391, 7.973566175466936, 8.187787464561668, 8.778111585409567, 9.398787253145600, 9.949816856806844, 10.26115505119941, 10.93174804156257, 11.44369050298382, 11.73761389920870, 11.88387178891786, 12.79936644921481, 12.92894208069401

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