Properties

Label 2-119952-1.1-c1-0-12
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 − 6·11-s − 5·13-s − 17-s + 2·19-s + 6·23-s + 4·25-s + 9·29-s − 7·31-s + 2·37-s + 9·41-s − 8·43-s − 3·47-s + 18·55-s − 3·59-s − 8·61-s + 15·65-s − 8·67-s + 12·71-s + 10·73-s + 16·79-s + 15·83-s + 3·85-s − 18·89-s − 6·95-s − 2·97-s + 101-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.80·11-s − 1.38·13-s − 0.242·17-s + 0.458·19-s + 1.25·23-s + 4/5·25-s + 1.67·29-s − 1.25·31-s + 0.328·37-s + 1.40·41-s − 1.21·43-s − 0.437·47-s + 2.42·55-s − 0.390·59-s − 1.02·61-s + 1.86·65-s − 0.977·67-s + 1.42·71-s + 1.17·73-s + 1.80·79-s + 1.64·83-s + 0.325·85-s − 1.90·89-s − 0.615·95-s − 0.203·97-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: \(0\)
Selberg data: \((2,\ 119952,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5566893834\)
\(L(\frac12)\) \(\approx\) \(0.5566893834\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.62647733628390, −12.90123397315896, −12.50607751462060, −12.27369810369965, −11.64257195297803, −11.07280695639430, −10.75063422628665, −10.32653081214006, −9.511415509940853, −9.347471082373641, −8.356936421866860, −8.143634833653461, −7.612780364006764, −7.257892481671883, −6.806743800100602, −6.018535047845176, −5.211779376716356, −4.900908245347443, −4.592736560258587, −3.728817965195243, −3.134023116503916, −2.690231243897016, −2.139595208693322, −0.9978015343629200, −0.2639835635299397, 0.2639835635299397, 0.9978015343629200, 2.139595208693322, 2.690231243897016, 3.134023116503916, 3.728817965195243, 4.592736560258587, 4.900908245347443, 5.211779376716356, 6.018535047845176, 6.806743800100602, 7.257892481671883, 7.612780364006764, 8.143634833653461, 8.356936421866860, 9.347471082373641, 9.511415509940853, 10.32653081214006, 10.75063422628665, 11.07280695639430, 11.64257195297803, 12.27369810369965, 12.50607751462060, 12.90123397315896, 13.62647733628390

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