Properties

Label 2-168e2-1.1-c1-0-118
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  + 2·11-s + 13-s − 2·17-s − 5·19-s + 6·23-s − 5·25-s − 8·29-s − 3·31-s + 9·37-s + 2·41-s + 43-s − 8·47-s + 6·53-s + 6·59-s − 2·61-s − 5·67-s + 4·71-s + 11·73-s + 5·79-s + 12·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.603·11-s + 0.277·13-s − 0.485·17-s − 1.14·19-s + 1.25·23-s − 25-s − 1.48·29-s − 0.538·31-s + 1.47·37-s + 0.312·41-s + 0.152·43-s − 1.16·47-s + 0.824·53-s + 0.781·59-s − 0.256·61-s − 0.610·67-s + 0.474·71-s + 1.28·73-s + 0.562·79-s + 1.27·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

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

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: $\chi_{28224} (1, \cdot )$
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 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.27461366881622, −14.89978477444844, −14.67308441509713, −13.77206884426463, −13.32577420580496, −12.86940342700691, −12.39641269013731, −11.57079373458958, −11.13268989620398, −10.88733248134280, −9.962812018020854, −9.489774563951798, −8.943883828836844, −8.476052753050616, −7.707813119711358, −7.239736861846936, −6.459979175381346, −6.121191381382610, −5.350188122332554, −4.677256013992926, −3.944023770474007, −3.569837775823854, −2.518616494510712, −1.948514839168169, −1.051617437731746, 0, 1.051617437731746, 1.948514839168169, 2.518616494510712, 3.569837775823854, 3.944023770474007, 4.677256013992926, 5.350188122332554, 6.121191381382610, 6.459979175381346, 7.239736861846936, 7.707813119711358, 8.476052753050616, 8.943883828836844, 9.489774563951798, 9.962812018020854, 10.88733248134280, 11.13268989620398, 11.57079373458958, 12.39641269013731, 12.86940342700691, 13.32577420580496, 13.77206884426463, 14.67308441509713, 14.89978477444844, 15.27461366881622

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