Properties

Label 2-159936-1.1-c1-0-100
Degree $2$
Conductor $159936$
Sign $-1$
Analytic cond. $1277.09$
Root an. cond. $35.7364$
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  + 3-s − 3·5-s + 9-s − 11-s − 5·13-s − 3·15-s − 17-s − 7·19-s + 23-s + 4·25-s + 27-s − 2·29-s + 6·31-s − 33-s − 8·37-s − 5·39-s − 7·41-s + 43-s − 3·45-s + 6·47-s − 51-s + 2·53-s + 3·55-s − 7·57-s − 10·59-s + 8·61-s + 15·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.301·11-s − 1.38·13-s − 0.774·15-s − 0.242·17-s − 1.60·19-s + 0.208·23-s + 4/5·25-s + 0.192·27-s − 0.371·29-s + 1.07·31-s − 0.174·33-s − 1.31·37-s − 0.800·39-s − 1.09·41-s + 0.152·43-s − 0.447·45-s + 0.875·47-s − 0.140·51-s + 0.274·53-s + 0.404·55-s − 0.927·57-s − 1.30·59-s + 1.02·61-s + 1.86·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(159936\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(1277.09\)
Root analytic conductor: \(35.7364\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{159936} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 159936,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−13.69873324348355, −12.80725175889168, −12.56027059325384, −12.20369574277424, −11.71098410905660, −11.08200118058712, −10.73904174736832, −10.16414336970226, −9.706912767035675, −9.132450545678191, −8.514772698206953, −8.206552124622477, −7.843605494559326, −7.220207221025796, −6.785320150006558, −6.459427508678017, −5.357940471089486, −5.059924221854949, −4.381621160930306, −3.983893218310822, −3.522614396774985, −2.717631772597047, −2.372352427670733, −1.675380575251550, −0.5991307447547378, 0, 0.5991307447547378, 1.675380575251550, 2.372352427670733, 2.717631772597047, 3.522614396774985, 3.983893218310822, 4.381621160930306, 5.059924221854949, 5.357940471089486, 6.459427508678017, 6.785320150006558, 7.220207221025796, 7.843605494559326, 8.206552124622477, 8.514772698206953, 9.132450545678191, 9.706912767035675, 10.16414336970226, 10.73904174736832, 11.08200118058712, 11.71098410905660, 12.20369574277424, 12.56027059325384, 12.80725175889168, 13.69873324348355

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