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Properties

Label 2-119952-1.1-c1-0-101
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·11-s + 6·13-s + 17-s − 2·19-s + 2·23-s − 25-s + 6·29-s − 4·31-s + 2·37-s + 6·41-s − 4·43-s + 8·47-s + 12·53-s + 4·55-s + 4·59-s + 8·61-s + 12·65-s + 4·67-s − 10·71-s − 8·73-s + 16·79-s − 12·83-s + 2·85-s + 6·89-s − 4·95-s − 4·97-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.603·11-s + 1.66·13-s + 0.242·17-s − 0.458·19-s + 0.417·23-s − 1/5·25-s + 1.11·29-s − 0.718·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 1.16·47-s + 1.64·53-s + 0.539·55-s + 0.520·59-s + 1.02·61-s + 1.48·65-s + 0.488·67-s − 1.18·71-s − 0.936·73-s + 1.80·79-s − 1.31·83-s + 0.216·85-s + 0.635·89-s − 0.410·95-s − 0.406·97-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 119952,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.573805585\)
\(L(\frac12)\) \(\approx\) \(4.573805585\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.71690392426977, −13.05247137621616, −12.81942173277591, −12.06957322223634, −11.60716102788141, −11.15175251425601, −10.54951563884944, −10.26998530091037, −9.628580677867291, −9.151271705899970, −8.575452639184058, −8.461053788932714, −7.551097834210208, −7.045802877950512, −6.425722516888621, −6.048912480545470, −5.647416672449534, −5.055776517414917, −4.202634091339997, −3.895945888603953, −3.222174925224406, −2.490019920027395, −1.914155876763334, −1.202821053937555, −0.7244766419993623, 0.7244766419993623, 1.202821053937555, 1.914155876763334, 2.490019920027395, 3.222174925224406, 3.895945888603953, 4.202634091339997, 5.055776517414917, 5.647416672449534, 6.048912480545470, 6.425722516888621, 7.045802877950512, 7.551097834210208, 8.461053788932714, 8.575452639184058, 9.151271705899970, 9.628580677867291, 10.26998530091037, 10.54951563884944, 11.15175251425601, 11.60716102788141, 12.06957322223634, 12.81942173277591, 13.05247137621616, 13.71690392426977

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