Properties

Label 2-159936-1.1-c1-0-246
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 9-s − 6·13-s + 2·15-s − 17-s − 8·23-s − 25-s − 27-s + 6·29-s + 8·31-s − 10·37-s + 6·39-s + 6·41-s − 12·43-s − 2·45-s + 51-s + 10·53-s − 8·59-s + 6·61-s + 12·65-s − 12·67-s + 8·69-s + 6·73-s + 75-s − 8·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.66·13-s + 0.516·15-s − 0.242·17-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 1.64·37-s + 0.960·39-s + 0.937·41-s − 1.82·43-s − 0.298·45-s + 0.140·51-s + 1.37·53-s − 1.04·59-s + 0.768·61-s + 1.48·65-s − 1.46·67-s + 0.963·69-s + 0.702·73-s + 0.115·75-s − 0.900·79-s + 1/9·81-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: \(2\)
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−13.64268522619822, −13.40994954191221, −12.43612555847567, −12.23307315591990, −11.84028565970362, −11.71851086440416, −10.89111878703008, −10.29285495867940, −10.12070362118978, −9.604762229717275, −8.928823833598951, −8.278658196956867, −7.938692868881994, −7.514461842013569, −6.782619755334819, −6.636241824957258, −5.865489147588900, −5.265180034669212, −4.806223760708030, −4.287887835215750, −3.890186507930304, −3.108226669908537, −2.498679547737763, −1.907111429567030, −1.053833915001922, 0, 0, 1.053833915001922, 1.907111429567030, 2.498679547737763, 3.108226669908537, 3.890186507930304, 4.287887835215750, 4.806223760708030, 5.265180034669212, 5.865489147588900, 6.636241824957258, 6.782619755334819, 7.514461842013569, 7.938692868881994, 8.278658196956867, 8.928823833598951, 9.604762229717275, 10.12070362118978, 10.29285495867940, 10.89111878703008, 11.71851086440416, 11.84028565970362, 12.23307315591990, 12.43612555847567, 13.40994954191221, 13.64268522619822

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