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Properties

Label 2-119952-1.1-c1-0-130
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  + 4·11-s − 5·13-s − 17-s + 7·19-s − 6·23-s − 5·25-s − 6·29-s + 31-s + 5·37-s − 6·41-s + 43-s + 2·47-s − 6·53-s + 4·59-s + 10·61-s − 13·67-s + 14·71-s − 11·73-s + 5·79-s + 2·83-s − 8·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s − 1.38·13-s − 0.242·17-s + 1.60·19-s − 1.25·23-s − 25-s − 1.11·29-s + 0.179·31-s + 0.821·37-s − 0.937·41-s + 0.152·43-s + 0.291·47-s − 0.824·53-s + 0.520·59-s + 1.28·61-s − 1.58·67-s + 1.66·71-s − 1.28·73-s + 0.562·79-s + 0.219·83-s − 0.847·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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: \(1\)
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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−13.82507643106137, −13.47570113975253, −12.76000293389678, −12.25371196548270, −11.77159435451687, −11.62748809249623, −11.03966313559611, −10.23281488601084, −9.775200707210701, −9.515874718572891, −9.124786902088830, −8.266870898701302, −7.903468282101659, −7.278418734913239, −6.990938979325646, −6.296172749538696, −5.685400366970636, −5.356133902279767, −4.537619541580132, −4.143661088404225, −3.519809263806295, −2.953326839278564, −2.113607893018373, −1.720632805234271, −0.8287597800205822, 0, 0.8287597800205822, 1.720632805234271, 2.113607893018373, 2.953326839278564, 3.519809263806295, 4.143661088404225, 4.537619541580132, 5.356133902279767, 5.685400366970636, 6.296172749538696, 6.990938979325646, 7.278418734913239, 7.903468282101659, 8.266870898701302, 9.124786902088830, 9.515874718572891, 9.775200707210701, 10.23281488601084, 11.03966313559611, 11.62748809249623, 11.77159435451687, 12.25371196548270, 12.76000293389678, 13.47570113975253, 13.82507643106137

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