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Properties

Label 2-141120-1.1-c1-0-337
Degree $2$
Conductor $141120$
Sign $-1$
Analytic cond. $1126.84$
Root an. cond. $33.5685$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·13-s − 6·17-s + 8·19-s + 25-s + 6·29-s + 4·31-s + 10·37-s − 6·41-s + 4·43-s − 6·53-s + 12·59-s − 10·61-s − 2·65-s + 4·67-s − 12·71-s + 10·73-s + 8·79-s − 12·83-s + 6·85-s − 6·89-s − 8·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.554·13-s − 1.45·17-s + 1.83·19-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 1.64·37-s − 0.937·41-s + 0.609·43-s − 0.824·53-s + 1.56·59-s − 1.28·61-s − 0.248·65-s + 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s − 1.31·83-s + 0.650·85-s − 0.635·89-s − 0.820·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141120\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(1126.84\)
Root analytic conductor: \(33.5685\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 141120,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.58851308708266, −13.29434772072344, −12.69404646421686, −12.06768610426181, −11.78084104358299, −11.12381348890096, −11.04552617772215, −10.22569040656459, −9.788754935904313, −9.289859081543992, −8.772657149508733, −8.253102245875947, −7.863607622613824, −7.235733826291162, −6.783414273505216, −6.244048359953875, −5.773242561226314, −4.964807584200133, −4.670981840728586, −4.022179893527174, −3.479092058859460, −2.793129970939100, −2.409420850744463, −1.357780533397959, −0.9302178152326234, 0, 0.9302178152326234, 1.357780533397959, 2.409420850744463, 2.793129970939100, 3.479092058859460, 4.022179893527174, 4.670981840728586, 4.964807584200133, 5.773242561226314, 6.244048359953875, 6.783414273505216, 7.235733826291162, 7.863607622613824, 8.253102245875947, 8.772657149508733, 9.289859081543992, 9.788754935904313, 10.22569040656459, 11.04552617772215, 11.12381348890096, 11.78084104358299, 12.06768610426181, 12.69404646421686, 13.29434772072344, 13.58851308708266

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