Properties

Label 2-119952-1.1-c1-0-127
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  − 2·5-s + 17-s + 2·19-s + 4·23-s − 25-s − 6·29-s + 8·31-s + 2·37-s − 10·41-s + 4·43-s + 8·47-s − 8·53-s + 4·59-s − 6·61-s + 4·67-s + 12·71-s + 10·73-s + 4·79-s + 12·83-s − 2·85-s − 6·89-s − 4·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.242·17-s + 0.458·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 1.09·53-s + 0.520·59-s − 0.768·61-s + 0.488·67-s + 1.42·71-s + 1.17·73-s + 0.450·79-s + 1.31·83-s − 0.216·85-s − 0.635·89-s − 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

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

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

−13.79353795213903, −13.34198843518672, −12.82758338438904, −12.19516749310018, −11.94420980818498, −11.44916583650655, −10.90776567692231, −10.56615744855119, −9.847860716252344, −9.375408078032698, −8.994739553287027, −8.193857015899127, −7.901364872904389, −7.543622724235403, −6.678443499043390, −6.594258218221519, −5.641760442718006, −5.199504903870582, −4.688955887001083, −3.891406753430815, −3.690389751320643, −2.911633420093172, −2.380767327874189, −1.468405776844077, −0.8180080597540304, 0, 0.8180080597540304, 1.468405776844077, 2.380767327874189, 2.911633420093172, 3.690389751320643, 3.891406753430815, 4.688955887001083, 5.199504903870582, 5.641760442718006, 6.594258218221519, 6.678443499043390, 7.543622724235403, 7.901364872904389, 8.193857015899127, 8.994739553287027, 9.375408078032698, 9.847860716252344, 10.56615744855119, 10.90776567692231, 11.44916583650655, 11.94420980818498, 12.19516749310018, 12.82758338438904, 13.34198843518672, 13.79353795213903

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