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Properties

Label 2-168e2-1.1-c1-0-166
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 + 11-s + 4·13-s − 4·17-s + 8·23-s + 4·25-s − 7·29-s − 11·31-s − 4·37-s + 4·41-s − 2·43-s − 2·47-s − 11·53-s + 3·55-s − 7·59-s − 10·61-s + 12·65-s + 10·67-s + 6·71-s − 6·73-s − 11·79-s − 11·83-s − 12·85-s − 6·89-s + 7·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.301·11-s + 1.10·13-s − 0.970·17-s + 1.66·23-s + 4/5·25-s − 1.29·29-s − 1.97·31-s − 0.657·37-s + 0.624·41-s − 0.304·43-s − 0.291·47-s − 1.51·53-s + 0.404·55-s − 0.911·59-s − 1.28·61-s + 1.48·65-s + 1.22·67-s + 0.712·71-s − 0.702·73-s − 1.23·79-s − 1.20·83-s − 1.30·85-s − 0.635·89-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 - T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−15.52678187675779, −14.86390389511094, −14.31784517999833, −13.92022343072964, −13.23578225252949, −12.93191457369054, −12.62775072571019, −11.45339943005617, −11.13501777176296, −10.75444891247928, −10.05697826375737, −9.247948091145523, −9.134929055758498, −8.655961185175661, −7.679245767587642, −7.108733599551109, −6.445831362790403, −6.039541854019007, −5.382297624350448, −4.894824068828006, −3.973456298608537, −3.362929756163732, −2.575927940053931, −1.694409940374000, −1.391676018247986, 0, 1.391676018247986, 1.694409940374000, 2.575927940053931, 3.362929756163732, 3.973456298608537, 4.894824068828006, 5.382297624350448, 6.039541854019007, 6.445831362790403, 7.108733599551109, 7.679245767587642, 8.655961185175661, 9.134929055758498, 9.247948091145523, 10.05697826375737, 10.75444891247928, 11.13501777176296, 11.45339943005617, 12.62775072571019, 12.93191457369054, 13.23578225252949, 13.92022343072964, 14.31784517999833, 14.86390389511094, 15.52678187675779

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