Properties

Label 2-259182-1.1-c1-0-113
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 + 2·20-s − 4·23-s − 25-s − 6·26-s − 28-s − 6·29-s + 2·31-s + 32-s − 34-s − 2·35-s − 4·37-s + 2·40-s + 2·41-s − 6·43-s − 4·46-s + 12·47-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 − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.447·20-s − 0.834·23-s − 1/5·25-s − 1.17·26-s − 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s − 0.171·34-s − 0.338·35-s − 0.657·37-s + 0.316·40-s + 0.312·41-s − 0.914·43-s − 0.589·46-s + 1.75·47-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: $\chi_{259182} (1, \cdot )$
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 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.06011041005607, −12.59970445715035, −12.23248071631460, −11.78019738859838, −11.39453558760570, −10.54330164795347, −10.37613209923608, −9.846482918413845, −9.395495689245319, −9.108801436314298, −8.252810084547168, −7.851734232546654, −7.202140680669295, −6.886593648496115, −6.395625652306527, −5.758743550077554, −5.368785732783720, −5.106894065356155, −4.258955215555594, −3.937658237183842, −3.290176433120417, −2.476654996389250, −2.282326345220063, −1.792752313312219, −0.8100823813557862, 0, 0.8100823813557862, 1.792752313312219, 2.282326345220063, 2.476654996389250, 3.290176433120417, 3.937658237183842, 4.258955215555594, 5.106894065356155, 5.368785732783720, 5.758743550077554, 6.395625652306527, 6.886593648496115, 7.202140680669295, 7.851734232546654, 8.252810084547168, 9.108801436314298, 9.395495689245319, 9.846482918413845, 10.37613209923608, 10.54330164795347, 11.39453558760570, 11.78019738859838, 12.23248071631460, 12.59970445715035, 13.06011041005607

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