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Properties

Label 2-124128-1.1-c1-0-14
Degree $2$
Conductor $124128$
Sign $1$
Analytic cond. $991.167$
Root an. cond. $31.4828$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 2·7-s − 3·11-s + 4·13-s − 7·19-s − 23-s + 4·25-s + 5·29-s − 4·31-s + 6·35-s − 8·37-s + 10·41-s + 12·43-s − 6·47-s − 3·49-s + 3·53-s + 9·55-s − 11·59-s − 10·61-s − 12·65-s − 8·67-s + 12·71-s − 12·73-s + 6·77-s − 10·79-s − 16·83-s − 6·89-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.755·7-s − 0.904·11-s + 1.10·13-s − 1.60·19-s − 0.208·23-s + 4/5·25-s + 0.928·29-s − 0.718·31-s + 1.01·35-s − 1.31·37-s + 1.56·41-s + 1.82·43-s − 0.875·47-s − 3/7·49-s + 0.412·53-s + 1.21·55-s − 1.43·59-s − 1.28·61-s − 1.48·65-s − 0.977·67-s + 1.42·71-s − 1.40·73-s + 0.683·77-s − 1.12·79-s − 1.75·83-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124128\)    =    \(2^{5} \cdot 3^{2} \cdot 431\)
Sign: $1$
Analytic conductor: \(991.167\)
Root analytic conductor: \(31.4828\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 124128,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
431 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.08387044889092, −13.38508386326459, −12.94742396479690, −12.54986855389393, −12.24552145077851, −11.58598961802029, −10.94031884674887, −10.76645162356609, −10.36983309755698, −9.582352895389381, −9.019624591576564, −8.568399092446156, −8.106510436934296, −7.707216653281529, −7.088752037553172, −6.635957674547388, −5.943919532473099, −5.693957081985414, −4.698788883367274, −4.228221535482814, −3.930891872109214, −3.090412055855771, −2.863862275000490, −1.932281508172322, −1.112664312338531, 0, 0, 1.112664312338531, 1.932281508172322, 2.863862275000490, 3.090412055855771, 3.930891872109214, 4.228221535482814, 4.698788883367274, 5.693957081985414, 5.943919532473099, 6.635957674547388, 7.088752037553172, 7.707216653281529, 8.106510436934296, 8.568399092446156, 9.019624591576564, 9.582352895389381, 10.36983309755698, 10.76645162356609, 10.94031884674887, 11.58598961802029, 12.24552145077851, 12.54986855389393, 12.94742396479690, 13.38508386326459, 14.08387044889092

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