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Properties

Label 2-159936-1.1-c1-0-130
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 − 5-s + 9-s + 5·11-s + 13-s − 15-s + 17-s + 2·19-s + 6·23-s − 4·25-s + 27-s − 2·29-s + 8·31-s + 5·33-s − 37-s + 39-s − 8·41-s − 43-s − 45-s + 4·47-s + 51-s + 5·53-s − 5·55-s + 2·57-s − 4·59-s + 6·61-s − 65-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 + 0.458·19-s + 1.25·23-s − 4/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.870·33-s − 0.164·37-s + 0.160·39-s − 1.24·41-s − 0.152·43-s − 0.149·45-s + 0.583·47-s + 0.140·51-s + 0.686·53-s − 0.674·55-s + 0.264·57-s − 0.520·59-s + 0.768·61-s − 0.124·65-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

\(L(1)\) \(\approx\) \(4.407940397\)
\(L(\frac12)\) \(\approx\) \(4.407940397\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−13.45393025831288, −12.76857691308697, −12.28738146045135, −11.82568778336826, −11.46776460525059, −11.04138901021669, −10.30152766960709, −9.871431448590222, −9.385982983988142, −8.951217450346532, −8.452004380569072, −8.088001144297976, −7.363182506170335, −7.057207395303399, −6.451472642365649, −6.062892031172666, −5.226050031219705, −4.781404847000667, −4.120363204142816, −3.555530548591132, −3.380624983834584, −2.513696677369301, −1.870496292320471, −1.144830819275039, −0.6692309010826609, 0.6692309010826609, 1.144830819275039, 1.870496292320471, 2.513696677369301, 3.380624983834584, 3.555530548591132, 4.120363204142816, 4.781404847000667, 5.226050031219705, 6.062892031172666, 6.451472642365649, 7.057207395303399, 7.363182506170335, 8.088001144297976, 8.452004380569072, 8.951217450346532, 9.385982983988142, 9.871431448590222, 10.30152766960709, 11.04138901021669, 11.46776460525059, 11.82568778336826, 12.28738146045135, 12.76857691308697, 13.45393025831288

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