Properties

Label 2-119952-1.1-c1-0-11
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  − 5·13-s − 17-s − 19-s − 6·23-s − 5·25-s + 6·29-s + 5·31-s − 7·37-s − 6·41-s + 43-s + 6·47-s − 6·53-s + 10·61-s − 5·67-s − 6·71-s + 73-s + 79-s + 6·83-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.38·13-s − 0.242·17-s − 0.229·19-s − 1.25·23-s − 25-s + 1.11·29-s + 0.898·31-s − 1.15·37-s − 0.937·41-s + 0.152·43-s + 0.875·47-s − 0.824·53-s + 1.28·61-s − 0.610·67-s − 0.712·71-s + 0.117·73-s + 0.112·79-s + 0.658·83-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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

\(L(1)\) \(\approx\) \(0.6840613356\)
\(L(\frac12)\) \(\approx\) \(0.6840613356\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.63132977540401, −13.13870030537707, −12.42666326290183, −12.05287526554428, −11.87113129429431, −11.21368333835274, −10.43069349261912, −10.20480136963220, −9.779982496922967, −9.198854230846640, −8.632937470112733, −8.032318510950471, −7.774191787413294, −7.022248469487476, −6.641305439075312, −6.082036896011394, −5.419180134424858, −4.973321915384123, −4.338178773636737, −3.923458108557542, −3.134737515756740, −2.492269208499065, −2.052298089960096, −1.282338382371784, −0.2487168106291979, 0.2487168106291979, 1.282338382371784, 2.052298089960096, 2.492269208499065, 3.134737515756740, 3.923458108557542, 4.338178773636737, 4.973321915384123, 5.419180134424858, 6.082036896011394, 6.641305439075312, 7.022248469487476, 7.774191787413294, 8.032318510950471, 8.632937470112733, 9.198854230846640, 9.779982496922967, 10.20480136963220, 10.43069349261912, 11.21368333835274, 11.87113129429431, 12.05287526554428, 12.42666326290183, 13.13870030537707, 13.63132977540401

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