Properties

Label 2-119952-1.1-c1-0-112
Degree $2$
Conductor $119952$
Sign $-1$
Analytic cond. $957.821$
Root an. cond. $30.9486$
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 − 2·11-s + 13-s − 17-s − 4·19-s + 8·23-s − 4·25-s − 3·29-s + 3·31-s + 4·37-s − 41-s + 4·43-s − 9·47-s + 2·55-s + 11·59-s − 65-s + 12·67-s + 4·71-s + 6·73-s + 9·83-s + 85-s + 6·89-s + 4·95-s + 8·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.603·11-s + 0.277·13-s − 0.242·17-s − 0.917·19-s + 1.66·23-s − 4/5·25-s − 0.557·29-s + 0.538·31-s + 0.657·37-s − 0.156·41-s + 0.609·43-s − 1.31·47-s + 0.269·55-s + 1.43·59-s − 0.124·65-s + 1.46·67-s + 0.474·71-s + 0.702·73-s + 0.987·83-s + 0.108·85-s + 0.635·89-s + 0.410·95-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119952\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(957.821\)
Root analytic conductor: \(30.9486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{119952} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 119952,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.60090803328191, −13.24674170013193, −12.99527322904772, −12.36241595377429, −11.89154467925374, −11.24435736122943, −10.97283936326020, −10.59165652429476, −9.805862120687622, −9.489981108354871, −8.861459595465693, −8.304419074075001, −7.966528057300578, −7.448885942975876, −6.638282182450475, −6.578407867338779, −5.695565265983388, −5.147506114343040, −4.760046693050552, −3.857986606322921, −3.764500643737669, −2.734906755208392, −2.439128383384931, −1.559884385288574, −0.7831449537514705, 0, 0.7831449537514705, 1.559884385288574, 2.439128383384931, 2.734906755208392, 3.764500643737669, 3.857986606322921, 4.760046693050552, 5.147506114343040, 5.695565265983388, 6.578407867338779, 6.638282182450475, 7.448885942975876, 7.966528057300578, 8.304419074075001, 8.861459595465693, 9.489981108354871, 9.805862120687622, 10.59165652429476, 10.97283936326020, 11.24435736122943, 11.89154467925374, 12.36241595377429, 12.99527322904772, 13.24674170013193, 13.60090803328191

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