Properties

Label 2-119952-1.1-c1-0-0
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 − 13-s − 17-s + 3·19-s − 5·23-s + 4·25-s + 2·29-s + 2·37-s + 7·41-s − 3·43-s − 12·47-s − 8·53-s + 15·55-s + 8·59-s − 6·61-s + 3·65-s + 8·67-s + 2·73-s − 12·79-s − 10·83-s + 3·85-s − 9·95-s − 12·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.50·11-s − 0.277·13-s − 0.242·17-s + 0.688·19-s − 1.04·23-s + 4/5·25-s + 0.371·29-s + 0.328·37-s + 1.09·41-s − 0.457·43-s − 1.75·47-s − 1.09·53-s + 2.02·55-s + 1.04·59-s − 0.768·61-s + 0.372·65-s + 0.977·67-s + 0.234·73-s − 1.35·79-s − 1.09·83-s + 0.325·85-s − 0.923·95-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\(L(1)\) \(\approx\) \(0.1052188265\)
\(L(\frac12)\) \(\approx\) \(0.1052188265\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−13.50361030099161, −12.95366122019002, −12.64037568579563, −12.10208214841095, −11.54941072172117, −11.27804269481024, −10.74649766990404, −10.12493375032201, −9.792405956064584, −9.177009373892842, −8.324465640374803, −8.142094396991050, −7.727218776054846, −7.270542272723035, −6.672289402487935, −6.033794917451022, −5.344185351938318, −4.958015293525451, −4.303555535899455, −3.893403436840747, −3.104863837675942, −2.771293152494480, −2.005473743234097, −1.111452306906678, −0.1068577563352411, 0.1068577563352411, 1.111452306906678, 2.005473743234097, 2.771293152494480, 3.104863837675942, 3.893403436840747, 4.303555535899455, 4.958015293525451, 5.344185351938318, 6.033794917451022, 6.672289402487935, 7.270542272723035, 7.727218776054846, 8.142094396991050, 8.324465640374803, 9.177009373892842, 9.792405956064584, 10.12493375032201, 10.74649766990404, 11.27804269481024, 11.54941072172117, 12.10208214841095, 12.64037568579563, 12.95366122019002, 13.50361030099161

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