Properties

Label 2-159936-1.1-c1-0-113
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 + 2·5-s + 9-s + 2·13-s + 2·15-s − 17-s + 2·23-s − 25-s + 27-s + 6·29-s + 8·37-s + 2·39-s − 6·41-s + 4·43-s + 2·45-s − 4·47-s − 51-s − 6·53-s + 6·59-s − 2·61-s + 4·65-s + 8·67-s + 2·69-s − 6·71-s + 6·73-s − 75-s + 4·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 0.242·17-s + 0.417·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.31·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.298·45-s − 0.583·47-s − 0.140·51-s − 0.824·53-s + 0.781·59-s − 0.256·61-s + 0.496·65-s + 0.977·67-s + 0.240·69-s − 0.712·71-s + 0.702·73-s − 0.115·75-s + 0.450·79-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: \(0\)
Selberg data: \((2,\ 159936,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.875346639\)
\(L(\frac12)\) \(\approx\) \(4.875346639\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.37413922134192, −12.83002590802534, −12.59233082399410, −11.78110577162532, −11.38465343040509, −10.87359912870164, −10.23124475371128, −9.974452470855108, −9.418463954094688, −8.991649760464212, −8.515647606611399, −8.017690419202523, −7.533204163878531, −6.869823183402604, −6.309773556301571, −6.093972477946989, −5.320953478430099, −4.812238656349305, −4.298339219174530, −3.559135675053930, −3.125588880684785, −2.392257751130634, −2.009544933106435, −1.269832343181073, −0.6398729290967218, 0.6398729290967218, 1.269832343181073, 2.009544933106435, 2.392257751130634, 3.125588880684785, 3.559135675053930, 4.298339219174530, 4.812238656349305, 5.320953478430099, 6.093972477946989, 6.309773556301571, 6.869823183402604, 7.533204163878531, 8.017690419202523, 8.515647606611399, 8.991649760464212, 9.418463954094688, 9.974452470855108, 10.23124475371128, 10.87359912870164, 11.38465343040509, 11.78110577162532, 12.59233082399410, 12.83002590802534, 13.37413922134192

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