Properties

Label 2-159936-1.1-c1-0-118
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 + 5-s + 9-s − 5·11-s − 13-s − 15-s + 17-s − 6·19-s + 6·23-s − 4·25-s − 27-s − 6·29-s − 4·31-s + 5·33-s − 11·37-s + 39-s + 9·43-s + 45-s − 4·47-s − 51-s + 7·53-s − 5·55-s + 6·57-s + 12·59-s + 6·61-s − 65-s − 13·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s − 1.37·19-s + 1.25·23-s − 4/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.870·33-s − 1.80·37-s + 0.160·39-s + 1.37·43-s + 0.149·45-s − 0.583·47-s − 0.140·51-s + 0.961·53-s − 0.674·55-s + 0.794·57-s + 1.56·59-s + 0.768·61-s − 0.124·65-s − 1.58·67-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 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−13.40659520059449, −13.00772994666550, −12.58979904812569, −12.27278285491898, −11.49635191167304, −11.05030297689818, −10.68990624751132, −10.22853969163629, −9.901779857334978, −9.078580977899776, −8.896582767753455, −8.105258522021002, −7.670075062895719, −7.161069834063628, −6.703839009797245, −6.041845754861343, −5.568686736460087, −5.106390474755951, −4.856391745064166, −3.834680211285338, −3.618555032087914, −2.551588085740350, −2.264702561053584, −1.629291009266562, −0.6479106679518939, 0, 0.6479106679518939, 1.629291009266562, 2.264702561053584, 2.551588085740350, 3.618555032087914, 3.834680211285338, 4.856391745064166, 5.106390474755951, 5.568686736460087, 6.041845754861343, 6.703839009797245, 7.161069834063628, 7.670075062895719, 8.105258522021002, 8.896582767753455, 9.078580977899776, 9.901779857334978, 10.22853969163629, 10.68990624751132, 11.05030297689818, 11.49635191167304, 12.27278285491898, 12.58979904812569, 13.00772994666550, 13.40659520059449

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