Properties

Label 2-119952-1.1-c1-0-104
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 − 13-s − 17-s − 4·19-s + 4·23-s + 4·25-s + 7·29-s + 7·31-s + 8·37-s + 3·41-s + 8·43-s − 7·47-s + 4·53-s + 18·55-s + 5·59-s − 4·61-s − 3·65-s − 4·67-s − 16·71-s + 2·73-s + 8·79-s − 9·83-s − 3·85-s − 14·89-s − 12·95-s − 8·97-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.80·11-s − 0.277·13-s − 0.242·17-s − 0.917·19-s + 0.834·23-s + 4/5·25-s + 1.29·29-s + 1.25·31-s + 1.31·37-s + 0.468·41-s + 1.21·43-s − 1.02·47-s + 0.549·53-s + 2.42·55-s + 0.650·59-s − 0.512·61-s − 0.372·65-s − 0.488·67-s − 1.89·71-s + 0.234·73-s + 0.900·79-s − 0.987·83-s − 0.325·85-s − 1.48·89-s − 1.23·95-s − 0.812·97-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

\(L(1)\) \(\approx\) \(4.958642112\)
\(L(\frac12)\) \(\approx\) \(4.958642112\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.62638344581346, −13.09480254201785, −12.70967324162264, −12.09738356145148, −11.66952208218192, −11.16935619548199, −10.55635841456649, −10.12096422827008, −9.525619523080880, −9.356923404710987, −8.627948780817459, −8.453017596690922, −7.513348306525506, −6.921780889609661, −6.471926634363692, −6.099822074040440, −5.733887895875568, −4.775738306712654, −4.476952528887379, −3.940944054789254, −2.978905921846106, −2.597790752275525, −1.883013998506076, −1.260938369296082, −0.7331957698927843, 0.7331957698927843, 1.260938369296082, 1.883013998506076, 2.597790752275525, 2.978905921846106, 3.940944054789254, 4.476952528887379, 4.775738306712654, 5.733887895875568, 6.099822074040440, 6.471926634363692, 6.921780889609661, 7.513348306525506, 8.453017596690922, 8.627948780817459, 9.356923404710987, 9.525619523080880, 10.12096422827008, 10.55635841456649, 11.16935619548199, 11.66952208218192, 12.09738356145148, 12.70967324162264, 13.09480254201785, 13.62638344581346

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