Properties

Label 2-164934-1.1-c1-0-108
Degree $2$
Conductor $164934$
Sign $-1$
Analytic cond. $1317.00$
Root an. cond. $36.2905$
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 + 8-s + 2·10-s − 11-s + 6·13-s + 16-s + 17-s + 2·20-s − 22-s − 2·23-s − 25-s + 6·26-s − 3·29-s − 2·31-s + 32-s + 34-s − 4·37-s + 2·40-s + 7·41-s + 12·43-s − 44-s − 2·46-s − 9·47-s − 50-s + 6·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 0.301·11-s + 1.66·13-s + 1/4·16-s + 0.242·17-s + 0.447·20-s − 0.213·22-s − 0.417·23-s − 1/5·25-s + 1.17·26-s − 0.557·29-s − 0.359·31-s + 0.176·32-s + 0.171·34-s − 0.657·37-s + 0.316·40-s + 1.09·41-s + 1.82·43-s − 0.150·44-s − 0.294·46-s − 1.31·47-s − 0.141·50-s + 0.832·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164934 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164934 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164934\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(1317.00\)
Root analytic conductor: \(36.2905\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 164934,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 + T \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.52159539100099, −13.14972810210521, −12.62647635008283, −12.24041868888542, −11.49006060984163, −11.18300443648777, −10.73188619204441, −10.18827144145675, −9.807379825854505, −9.113829376438899, −8.785659811856494, −8.115688265136015, −7.676392481286509, −7.012883707987705, −6.549537252656273, −5.861308303605453, −5.715057170813756, −5.324986924720296, −4.407520451291340, −3.959219488686701, −3.554638605468228, −2.711547567583492, −2.357177227033540, −1.477860035193911, −1.210678604867413, 0, 1.210678604867413, 1.477860035193911, 2.357177227033540, 2.711547567583492, 3.554638605468228, 3.959219488686701, 4.407520451291340, 5.324986924720296, 5.715057170813756, 5.861308303605453, 6.549537252656273, 7.012883707987705, 7.676392481286509, 8.115688265136015, 8.785659811856494, 9.113829376438899, 9.807379825854505, 10.18827144145675, 10.73188619204441, 11.18300443648777, 11.49006060984163, 12.24041868888542, 12.62647635008283, 13.14972810210521, 13.52159539100099

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