Properties

Label 2-168e2-1.1-c1-0-113
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  − 4·13-s + 6·17-s + 2·19-s − 5·25-s − 6·29-s + 4·31-s − 2·37-s + 6·41-s − 8·43-s − 12·47-s + 6·53-s + 6·59-s + 8·61-s + 4·67-s − 2·73-s + 8·79-s + 6·83-s − 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.10·13-s + 1.45·17-s + 0.458·19-s − 25-s − 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 0.824·53-s + 0.781·59-s + 1.02·61-s + 0.488·67-s − 0.234·73-s + 0.900·79-s + 0.658·83-s − 0.635·89-s + 1.01·97-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 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.35231965333849, −14.96976260059801, −14.29482422502816, −14.14204513626903, −13.17970777485948, −12.95530419283022, −12.16275059698363, −11.72653549176093, −11.42453006806420, −10.47349085550044, −9.941839656954204, −9.707432703860519, −9.040694804518053, −8.180875231063652, −7.801622845514357, −7.284696382845970, −6.625375357727742, −5.902618453702980, −5.254970731200679, −4.919080375737291, −3.897101947701589, −3.457023811465238, −2.625166654235116, −1.923349151904354, −1.030370765192138, 0, 1.030370765192138, 1.923349151904354, 2.625166654235116, 3.457023811465238, 3.897101947701589, 4.919080375737291, 5.254970731200679, 5.902618453702980, 6.625375357727742, 7.284696382845970, 7.801622845514357, 8.180875231063652, 9.040694804518053, 9.707432703860519, 9.941839656954204, 10.47349085550044, 11.42453006806420, 11.72653549176093, 12.16275059698363, 12.95530419283022, 13.17970777485948, 14.14204513626903, 14.29482422502816, 14.96976260059801, 15.35231965333849

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