Properties

Label 2-119952-1.1-c1-0-110
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 − 6·13-s − 17-s − 8·19-s − 2·23-s − 25-s + 6·29-s − 8·31-s − 4·37-s + 10·41-s − 8·43-s + 4·47-s + 10·53-s + 14·59-s + 6·61-s − 12·65-s + 4·67-s + 6·71-s − 10·73-s + 4·79-s + 6·83-s − 2·85-s + 6·89-s − 16·95-s + 18·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.66·13-s − 0.242·17-s − 1.83·19-s − 0.417·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.657·37-s + 1.56·41-s − 1.21·43-s + 0.583·47-s + 1.37·53-s + 1.82·59-s + 0.768·61-s − 1.48·65-s + 0.488·67-s + 0.712·71-s − 1.17·73-s + 0.450·79-s + 0.658·83-s − 0.216·85-s + 0.635·89-s − 1.64·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.86364777906926, −13.16811309137626, −12.94169324408101, −12.43682106242530, −11.91113822202791, −11.48720805449091, −10.66520032228657, −10.40122627092515, −9.963343796986512, −9.481961122946198, −8.916214447770285, −8.513404613562336, −7.893690901609555, −7.250349739506515, −6.851752354235263, −6.310555372262677, −5.756883477566619, −5.236053543717768, −4.736797077373348, −4.117085199647060, −3.590095345841213, −2.579348158341866, −2.207898046898368, −1.923623327984015, −0.7795190798636442, 0, 0.7795190798636442, 1.923623327984015, 2.207898046898368, 2.579348158341866, 3.590095345841213, 4.117085199647060, 4.736797077373348, 5.236053543717768, 5.756883477566619, 6.310555372262677, 6.851752354235263, 7.250349739506515, 7.893690901609555, 8.513404613562336, 8.916214447770285, 9.481961122946198, 9.963343796986512, 10.40122627092515, 10.66520032228657, 11.48720805449091, 11.91113822202791, 12.43682106242530, 12.94169324408101, 13.16811309137626, 13.86364777906926

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