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Properties

Label 2-159936-1.1-c1-0-125
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·5-s + 9-s + 3·11-s − 13-s + 3·15-s + 17-s + 4·19-s − 6·23-s + 4·25-s + 27-s − 4·29-s − 2·31-s + 3·33-s − 37-s − 39-s + 10·41-s − 43-s + 3·45-s + 6·47-s + 51-s + 7·53-s + 9·55-s + 4·57-s + 4·59-s − 14·61-s − 3·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.774·15-s + 0.242·17-s + 0.917·19-s − 1.25·23-s + 4/5·25-s + 0.192·27-s − 0.742·29-s − 0.359·31-s + 0.522·33-s − 0.164·37-s − 0.160·39-s + 1.56·41-s − 0.152·43-s + 0.447·45-s + 0.875·47-s + 0.140·51-s + 0.961·53-s + 1.21·55-s + 0.529·57-s + 0.520·59-s − 1.79·61-s − 0.372·65-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 159936,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.523939538\)
\(L(\frac12)\) \(\approx\) \(5.523939538\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.43382903731038, −12.80461851791993, −12.49930581532497, −11.81083459560276, −11.51275497436465, −10.77811678241580, −10.18506773220454, −9.946998804342069, −9.302883850820615, −9.193989001836727, −8.619800307996590, −7.911853062793199, −7.400982208895072, −7.060318466958997, −6.272231093780869, −5.848563352882414, −5.596249290345878, −4.781312543767703, −4.234943816821909, −3.623402188149252, −3.117211535358247, −2.290138411549452, −2.012843421141960, −1.352549278625097, −0.6609814164850233, 0.6609814164850233, 1.352549278625097, 2.012843421141960, 2.290138411549452, 3.117211535358247, 3.623402188149252, 4.234943816821909, 4.781312543767703, 5.596249290345878, 5.848563352882414, 6.272231093780869, 7.060318466958997, 7.400982208895072, 7.911853062793199, 8.619800307996590, 9.193989001836727, 9.302883850820615, 9.946998804342069, 10.18506773220454, 10.77811678241580, 11.51275497436465, 11.81083459560276, 12.49930581532497, 12.80461851791993, 13.43382903731038

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