Properties

Label 2-54096-1.1-c1-0-17
Degree $2$
Conductor $54096$
Sign $1$
Analytic cond. $431.958$
Root an. cond. $20.7836$
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-s + 2·5-s + 9-s − 4·11-s − 2·13-s + 2·15-s − 6·17-s − 23-s − 25-s + 27-s − 2·29-s + 4·31-s − 4·33-s + 6·37-s − 2·39-s + 6·41-s − 12·43-s + 2·45-s − 12·47-s − 6·51-s + 6·53-s − 8·55-s − 4·59-s + 10·61-s − 4·65-s − 4·67-s − 69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.516·15-s − 1.45·17-s − 0.208·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 1.82·43-s + 0.298·45-s − 1.75·47-s − 0.840·51-s + 0.824·53-s − 1.07·55-s − 0.520·59-s + 1.28·61-s − 0.496·65-s − 0.488·67-s − 0.120·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54096\)    =    \(2^{4} \cdot 3 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(431.958\)
Root analytic conductor: \(20.7836\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 54096,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.105911589\)
\(L(\frac12)\) \(\approx\) \(2.105911589\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
23 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
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 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.34513307654598, −13.92220680570510, −13.27410454567362, −13.05412356120601, −12.76475681932050, −11.75333503939918, −11.41605675755948, −10.71546970110879, −10.20284561386587, −9.668873926008237, −9.496531488838848, −8.540973266308804, −8.323058099306256, −7.668780641299802, −7.034615025163774, −6.522729075989260, −5.897709159453227, −5.295774672822322, −4.705143779594285, −4.213980060456619, −3.286975076242293, −2.644663417100188, −2.194841685945922, −1.652132286498091, −0.4496486900189149, 0.4496486900189149, 1.652132286498091, 2.194841685945922, 2.644663417100188, 3.286975076242293, 4.213980060456619, 4.705143779594285, 5.295774672822322, 5.897709159453227, 6.522729075989260, 7.034615025163774, 7.668780641299802, 8.323058099306256, 8.540973266308804, 9.496531488838848, 9.668873926008237, 10.20284561386587, 10.71546970110879, 11.41605675755948, 11.75333503939918, 12.76475681932050, 13.05412356120601, 13.27410454567362, 13.92220680570510, 14.34513307654598

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