Properties

Label 2-259182-1.1-c1-0-140
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 − 6·13-s + 14-s + 16-s + 17-s + 4·19-s + 2·20-s − 6·23-s − 25-s − 6·26-s + 28-s + 2·29-s + 4·31-s + 32-s + 34-s + 2·35-s + 6·37-s + 4·38-s + 2·40-s − 10·41-s − 8·43-s − 6·46-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 − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.447·20-s − 1.25·23-s − 1/5·25-s − 1.17·26-s + 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.176·32-s + 0.171·34-s + 0.338·35-s + 0.986·37-s + 0.648·38-s + 0.316·40-s − 1.56·41-s − 1.21·43-s − 0.884·46-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: $\chi_{259182} (1, \cdot )$
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 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−13.21719839153019, −12.58853455698274, −11.97898277797113, −11.79421121669813, −11.56853221668969, −10.60060321044003, −10.22487621031190, −9.846356790964828, −9.626138953935985, −8.918545900418546, −8.130561381840211, −7.984162978141786, −7.266138703983629, −6.901242009397210, −6.321460578471934, −5.817786491406810, −5.332749156198527, −4.951203168377393, −4.521726477055387, −3.840238840134886, −3.242947279531999, −2.615822088327888, −2.175598938855039, −1.703422673685938, −0.9445912662544611, 0, 0.9445912662544611, 1.703422673685938, 2.175598938855039, 2.615822088327888, 3.242947279531999, 3.840238840134886, 4.521726477055387, 4.951203168377393, 5.332749156198527, 5.817786491406810, 6.321460578471934, 6.901242009397210, 7.266138703983629, 7.984162978141786, 8.130561381840211, 8.918545900418546, 9.626138953935985, 9.846356790964828, 10.22487621031190, 10.60060321044003, 11.56853221668969, 11.79421121669813, 11.97898277797113, 12.58853455698274, 13.21719839153019

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