Properties

Label 2-119952-1.1-c1-0-161
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  + 3·5-s + 2·11-s − 17-s + 3·19-s − 5·23-s + 4·25-s − 6·29-s + 3·37-s + 43-s − 6·47-s + 6·53-s + 6·55-s + 5·59-s + 10·61-s + 7·67-s + 3·71-s − 4·73-s − 16·79-s − 12·83-s − 3·85-s − 5·89-s + 9·95-s − 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.603·11-s − 0.242·17-s + 0.688·19-s − 1.04·23-s + 4/5·25-s − 1.11·29-s + 0.493·37-s + 0.152·43-s − 0.875·47-s + 0.824·53-s + 0.809·55-s + 0.650·59-s + 1.28·61-s + 0.855·67-s + 0.356·71-s − 0.468·73-s − 1.80·79-s − 1.31·83-s − 0.325·85-s − 0.529·89-s + 0.923·95-s − 0.406·97-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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
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

−13.86700328355701, −13.30041208938674, −12.96119532001184, −12.49836680484438, −11.72953896381809, −11.46486316964324, −10.95830460777491, −10.11567971820216, −9.939355401277149, −9.563853398033641, −8.956184243909945, −8.532523875018140, −7.913918544573987, −7.219246994272780, −6.858545654504540, −6.141944636047955, −5.835053942435781, −5.360362615557877, −4.759290856078913, −3.981969563433301, −3.615909938970202, −2.697573872043626, −2.240170514736555, −1.614462115851535, −1.061657245140739, 0, 1.061657245140739, 1.614462115851535, 2.240170514736555, 2.697573872043626, 3.615909938970202, 3.981969563433301, 4.759290856078913, 5.360362615557877, 5.835053942435781, 6.141944636047955, 6.858545654504540, 7.219246994272780, 7.913918544573987, 8.532523875018140, 8.956184243909945, 9.563853398033641, 9.939355401277149, 10.11567971820216, 10.95830460777491, 11.46486316964324, 11.72953896381809, 12.49836680484438, 12.96119532001184, 13.30041208938674, 13.86700328355701

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