Properties

Label 2-159936-1.1-c1-0-109
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 + 3·11-s + 3·13-s + 15-s − 17-s − 3·19-s − 7·23-s − 4·25-s + 27-s + 6·29-s + 10·31-s + 3·33-s − 4·37-s + 3·39-s + 9·41-s + 9·43-s + 45-s + 6·47-s − 51-s + 10·53-s + 3·55-s − 3·57-s + 2·59-s + 3·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.904·11-s + 0.832·13-s + 0.258·15-s − 0.242·17-s − 0.688·19-s − 1.45·23-s − 4/5·25-s + 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.522·33-s − 0.657·37-s + 0.480·39-s + 1.40·41-s + 1.37·43-s + 0.149·45-s + 0.875·47-s − 0.140·51-s + 1.37·53-s + 0.404·55-s − 0.397·57-s + 0.260·59-s + 0.372·65-s − 1.46·67-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: \(0\)
Selberg data: \((2,\ 159936,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.597789263\)
\(L(\frac12)\) \(\approx\) \(4.597789263\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.43265932464047, −12.88157692302790, −12.25766081934639, −11.93534519017159, −11.49302540477524, −10.74791962751663, −10.36152242214744, −9.958458886996197, −9.390815710326702, −8.925559315115037, −8.486490436054431, −8.106852212686100, −7.500762791901544, −6.882683701310044, −6.341637179011173, −5.989005631396853, −5.568404916905754, −4.542629855613055, −4.105057959968490, −3.957624189716101, −2.990331725643391, −2.491371794419097, −1.937901066307973, −1.258003019013841, −0.6297540852175877, 0.6297540852175877, 1.258003019013841, 1.937901066307973, 2.491371794419097, 2.990331725643391, 3.957624189716101, 4.105057959968490, 4.542629855613055, 5.568404916905754, 5.989005631396853, 6.341637179011173, 6.882683701310044, 7.500762791901544, 8.106852212686100, 8.486490436054431, 8.925559315115037, 9.390815710326702, 9.958458886996197, 10.36152242214744, 10.74791962751663, 11.49302540477524, 11.93534519017159, 12.25766081934639, 12.88157692302790, 13.43265932464047

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