Properties

Label 2-141120-1.1-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 5·11-s − 3·13-s − 17-s + 6·19-s − 6·23-s + 25-s − 9·29-s + 4·31-s − 2·37-s − 4·41-s − 10·43-s − 47-s + 4·53-s − 5·55-s + 8·59-s − 8·61-s − 3·65-s − 12·67-s − 8·71-s − 2·73-s + 13·79-s + 4·83-s − 85-s + 4·89-s + 6·95-s + 13·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.50·11-s − 0.832·13-s − 0.242·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s − 1.67·29-s + 0.718·31-s − 0.328·37-s − 0.624·41-s − 1.52·43-s − 0.145·47-s + 0.549·53-s − 0.674·55-s + 1.04·59-s − 1.02·61-s − 0.372·65-s − 1.46·67-s − 0.949·71-s − 0.234·73-s + 1.46·79-s + 0.439·83-s − 0.108·85-s + 0.423·89-s + 0.615·95-s + 1.31·97-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: \(0\)
Selberg data: \((2,\ 141120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4800413114\)
\(L(\frac12)\) \(\approx\) \(0.4800413114\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−13.40706041135324, −13.21520176755191, −12.29789838318906, −12.04163301676510, −11.60046304849769, −10.92409405286572, −10.39261477216374, −10.06791797147396, −9.638083467769288, −9.155299091670351, −8.496134130219262, −7.879428169523519, −7.645659034613165, −7.086890344550483, −6.495064548399342, −5.797264507179394, −5.388556779577029, −5.024758625125080, −4.438247738164951, −3.620446242782462, −3.115244882017970, −2.474261628640342, −1.999775227258540, −1.312751941293158, −0.1976565167276836, 0.1976565167276836, 1.312751941293158, 1.999775227258540, 2.474261628640342, 3.115244882017970, 3.620446242782462, 4.438247738164951, 5.024758625125080, 5.388556779577029, 5.797264507179394, 6.495064548399342, 7.086890344550483, 7.645659034613165, 7.879428169523519, 8.496134130219262, 9.155299091670351, 9.638083467769288, 10.06791797147396, 10.39261477216374, 10.92409405286572, 11.60046304849769, 12.04163301676510, 12.29789838318906, 13.21520176755191, 13.40706041135324

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