Properties

Label 2-156e2-1.1-c1-0-8
Degree $2$
Conductor $24336$
Sign $1$
Analytic cond. $194.323$
Root an. cond. $13.9400$
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  − 3·5-s + 17-s + 4·23-s + 4·25-s − 3·29-s − 8·31-s − 5·37-s − 3·41-s − 4·43-s − 8·47-s − 7·49-s + 13·53-s + 12·59-s + 15·61-s − 12·67-s + 8·71-s + 3·73-s + 4·79-s + 12·83-s − 3·85-s − 10·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.242·17-s + 0.834·23-s + 4/5·25-s − 0.557·29-s − 1.43·31-s − 0.821·37-s − 0.468·41-s − 0.609·43-s − 1.16·47-s − 49-s + 1.78·53-s + 1.56·59-s + 1.92·61-s − 1.46·67-s + 0.949·71-s + 0.351·73-s + 0.450·79-s + 1.31·83-s − 0.325·85-s − 1.05·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24336\)    =    \(2^{4} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(194.323\)
Root analytic conductor: \(13.9400\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{24336} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 24336,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9527373100\)
\(L(\frac12)\) \(\approx\) \(0.9527373100\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 15 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.25996281166068, −14.85480319182078, −14.67507451320275, −13.71918420931452, −13.18354233621859, −12.69555304915686, −12.07480160251666, −11.59562991182869, −11.17642732920170, −10.62226859424140, −9.942263003408594, −9.297224531394511, −8.654876741416315, −8.157966631435856, −7.658375953630202, −6.849987462557047, −6.781281577185147, −5.418765280423558, −5.278731628002622, −4.311444811315399, −3.687124259763101, −3.339976390107249, −2.353930848060974, −1.429687377401806, −0.4001509970909771, 0.4001509970909771, 1.429687377401806, 2.353930848060974, 3.339976390107249, 3.687124259763101, 4.311444811315399, 5.278731628002622, 5.418765280423558, 6.781281577185147, 6.849987462557047, 7.658375953630202, 8.157966631435856, 8.654876741416315, 9.297224531394511, 9.942263003408594, 10.62226859424140, 11.17642732920170, 11.59562991182869, 12.07480160251666, 12.69555304915686, 13.18354233621859, 13.71918420931452, 14.67507451320275, 14.85480319182078, 15.25996281166068

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