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Properties

Label 2-168e2-1.1-c1-0-122
Degree $2$
Conductor $28224$
Sign $-1$
Analytic cond. $225.369$
Root an. cond. $15.0123$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 5·11-s + 2·13-s + 2·17-s + 6·19-s − 2·23-s + 4·25-s + 29-s − 31-s − 10·37-s + 4·41-s + 4·43-s − 8·47-s + 5·53-s − 15·55-s − 13·59-s − 8·61-s − 6·65-s − 14·67-s + 12·71-s + 6·73-s − 11·79-s + 7·83-s − 6·85-s − 6·89-s − 18·95-s − 19·97-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.50·11-s + 0.554·13-s + 0.485·17-s + 1.37·19-s − 0.417·23-s + 4/5·25-s + 0.185·29-s − 0.179·31-s − 1.64·37-s + 0.624·41-s + 0.609·43-s − 1.16·47-s + 0.686·53-s − 2.02·55-s − 1.69·59-s − 1.02·61-s − 0.744·65-s − 1.71·67-s + 1.42·71-s + 0.702·73-s − 1.23·79-s + 0.768·83-s − 0.650·85-s − 0.635·89-s − 1.84·95-s − 1.92·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(225.369\)
Root analytic conductor: \(15.0123\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−15.52665162389629, −14.96544878926149, −14.48559475473907, −13.76524258183969, −13.68219911441144, −12.46742145702510, −12.17663982759542, −11.89567274195197, −11.15455079871398, −10.93274465928591, −9.995600314772722, −9.460657655807459, −8.894839270487323, −8.359956385160759, −7.700079067100877, −7.309989375159726, −6.657734540922340, −6.038830007111869, −5.323406133275690, −4.544699470399184, −3.936063668143223, −3.501576529379924, −2.941203035285766, −1.616683743602458, −1.080014366947245, 0, 1.080014366947245, 1.616683743602458, 2.941203035285766, 3.501576529379924, 3.936063668143223, 4.544699470399184, 5.323406133275690, 6.038830007111869, 6.657734540922340, 7.309989375159726, 7.700079067100877, 8.359956385160759, 8.894839270487323, 9.460657655807459, 9.995600314772722, 10.93274465928591, 11.15455079871398, 11.89567274195197, 12.17663982759542, 12.46742145702510, 13.68219911441144, 13.76524258183969, 14.48559475473907, 14.96544878926149, 15.52665162389629

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