Properties

Label 2-119952-1.1-c1-0-133
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 − 5·11-s + 5·13-s + 17-s − 5·19-s − 23-s − 4·25-s + 6·29-s − 6·31-s + 4·37-s + 7·41-s + 7·43-s − 6·47-s − 6·53-s − 5·55-s − 14·59-s + 5·65-s + 12·67-s + 4·71-s − 6·73-s + 6·79-s + 6·83-s + 85-s + 12·89-s − 5·95-s − 8·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.50·11-s + 1.38·13-s + 0.242·17-s − 1.14·19-s − 0.208·23-s − 4/5·25-s + 1.11·29-s − 1.07·31-s + 0.657·37-s + 1.09·41-s + 1.06·43-s − 0.875·47-s − 0.824·53-s − 0.674·55-s − 1.82·59-s + 0.620·65-s + 1.46·67-s + 0.474·71-s − 0.702·73-s + 0.675·79-s + 0.658·83-s + 0.108·85-s + 1.27·89-s − 0.512·95-s − 0.812·97-s + 0.0995·101-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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−13.83207031047953, −13.20093047315392, −12.96892614346725, −12.50097942583070, −11.95596301043787, −11.13397587113659, −10.82995437735252, −10.61273653429552, −9.931254847680979, −9.390227501738128, −8.991978897300274, −8.189664911967058, −8.016742202755156, −7.561543036229204, −6.612967871179944, −6.336280050714707, −5.710905682327668, −5.409232382486616, −4.597970018842096, −4.160136378477316, −3.459088429298541, −2.843922215796519, −2.244307565268603, −1.686692116713192, −0.8436861846782433, 0, 0.8436861846782433, 1.686692116713192, 2.244307565268603, 2.843922215796519, 3.459088429298541, 4.160136378477316, 4.597970018842096, 5.409232382486616, 5.710905682327668, 6.336280050714707, 6.612967871179944, 7.561543036229204, 8.016742202755156, 8.189664911967058, 8.991978897300274, 9.390227501738128, 9.931254847680979, 10.61273653429552, 10.82995437735252, 11.13397587113659, 11.95596301043787, 12.50097942583070, 12.96892614346725, 13.20093047315392, 13.83207031047953

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