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Properties

Label 2-119952-1.1-c1-0-103
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 5·11-s + 3·13-s + 17-s − 5·19-s + 5·23-s + 4·25-s − 2·29-s + 8·31-s − 6·37-s + 9·41-s + 43-s − 4·53-s + 15·55-s − 12·59-s + 6·61-s + 9·65-s + 16·67-s + 8·71-s + 10·73-s − 8·79-s − 14·83-s + 3·85-s − 8·89-s − 15·95-s + 8·97-s + 101-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.50·11-s + 0.832·13-s + 0.242·17-s − 1.14·19-s + 1.04·23-s + 4/5·25-s − 0.371·29-s + 1.43·31-s − 0.986·37-s + 1.40·41-s + 0.152·43-s − 0.549·53-s + 2.02·55-s − 1.56·59-s + 0.768·61-s + 1.11·65-s + 1.95·67-s + 0.949·71-s + 1.17·73-s − 0.900·79-s − 1.53·83-s + 0.325·85-s − 0.847·89-s − 1.53·95-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 119952,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.057744621\)
\(L(\frac12)\) \(\approx\) \(5.057744621\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−13.64236409991422, −13.11507724059302, −12.58894556312037, −12.33266433456702, −11.48363520167798, −11.11513276154137, −10.73852419547058, −9.998016981445587, −9.697302491980053, −9.186185062296109, −8.712958234346381, −8.389162969317777, −7.595611597643839, −6.805226443231419, −6.557595743222657, −6.090779204994548, −5.659583146988806, −4.965150431508366, −4.370059719759254, −3.818087182899878, −3.196893715111173, −2.467589835107948, −1.865051329422158, −1.297836601887848, −0.7326381106430472, 0.7326381106430472, 1.297836601887848, 1.865051329422158, 2.467589835107948, 3.196893715111173, 3.818087182899878, 4.370059719759254, 4.965150431508366, 5.659583146988806, 6.090779204994548, 6.557595743222657, 6.805226443231419, 7.595611597643839, 8.389162969317777, 8.712958234346381, 9.186185062296109, 9.697302491980053, 9.998016981445587, 10.73852419547058, 11.11513276154137, 11.48363520167798, 12.33266433456702, 12.58894556312037, 13.11507724059302, 13.64236409991422

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